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Limits at infinity practice problems

End behaviour of rational and radical functions: divide through by the highest power. 62 problems with worked solutions; in 61 of them every equation is proved by a computer algebra system.

20 worked problems

Try each one before opening the solution. Every line is labeled with what checked it; 20 of these also have a second solution, written separately and checked the same way. How we verify

Problem 1.65

hard Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{4 \left(2 x - 1\right)^{2} - 1}{2 x + 2 \left(2 x - 1\right)^{2} - 1} \).
  1. \[ \lim_{x \to \infty}\left(\frac{4 \left(2 x - 1\right)^{2} - 1}{2 x + 2 \left(2 x - 1\right)^{2} - 1}\right) \]
    limitStarting with the original limit expression.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{16 x^{2} - 16 x + 3}{8 x^{2} - 6 x + 1}\right) \]
    algebra algebra simplifyExpand the squared term (2x - 1)^2. Distribute the constants into the parentheses. Combine like terms in the numerator and denominator.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{16 x^{2}}{8 x^{2} - 6 x + 1} - \frac{16 x}{8 x^{2} - 6 x + 1} + \frac{3}{8 x^{2} - 6 x + 1}\right) \]
    limit-lawSplit the limit into three separate parts using limit laws.✓ Proved
  4. \[ = - \lim_{x \to \infty}\left(\frac{16 x}{8 x^{2} - 6 x + 1}\right) + \lim_{x \to \infty}\left(\frac{16 x^{2}}{8 x^{2} - 6 x + 1}\right) + \lim_{x \to \infty}\left(\frac{3}{8 x^{2} - 6 x + 1}\right) \]
    limit-lawApply the limit to each term individually.✓ Proved
  5. \[ = \lim_{x \to \infty} 2 - \lim_{x \to \infty}\left(\frac{16 x}{8 x^{2} - 6 x + 1}\right) + \lim_{x \to \infty}\left(\frac{3}{8 x^{2} - 6 x + 1}\right) \]
    simplifyDivide the first term's numerator and denominator by x^2.✓ Proved
  6. \[ = 2 - \lim_{x \to \infty}\left(\frac{16 x}{8 x^{2} - 6 x + 1}\right) + \lim_{x \to \infty}\left(\frac{3}{8 x^{2} - 6 x + 1}\right) \]
    simplifyEvaluate the limit of the first term.✓ Proved
  7. \[ = 2 - \lim_{x \to \infty}\left(\frac{16}{8 x - 6 + \frac{1}{x}}\right) + \lim_{x \to \infty}\left(\frac{3}{8 x^{2} - 6 x + 1}\right) \]
    algebraDivide the second term's numerator and denominator by x.✓ Proved
  8. \[ = 2 + \lim_{x \to \infty}\left(\frac{3}{8 x^{2} - 6 x + 1}\right) \]
    limitEvaluate the limit of the second term as x approaches infinity.✓ Proved
  9. \[ = 2 \]
    limit simplifyEvaluate the limit of the third term as x approaches infinity. Final calculation.✓ Proved
Answer \( 2 \)
The graph of f(x) = (4(2x - 1)² - 1)/(2x + 2(2x - 1)² - 1). A point runs right along the curve; its height settles toward 2.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{4 \left(2 x - 1\right)^{2} - 1}{2 x + 2 \left(2 x - 1\right)^{2} - 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{16 x^{2} - 16 x + 3}{8 x^{2} - 6 x + 1}\right) \]
    rewrite algebra algebraExpand (2*x - 1)**2 Simplify the constants Combine like terms✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{16 - \frac{16}{x} + \frac{3}{x^{2}}}{8 - \frac{6}{x} + \frac{1}{x^{2}}}\right) \]
    simplifyDivide numerator and denominator by x**2✓ Proved
  4. \[ = 2 \]
    limitAs x→∞ the terms with 1/x vanish, giving 16/8 = 2✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.368

hard Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{3 \left(2 x + 1\right)^{2} + 1}{\left(2 x + 1\right)^{2} - 4} \).
  1. \[ \lim_{x \to \infty}\left(\frac{3 \left(2 x + 1\right)^{2} + 1}{\left(2 x + 1\right)^{2} - 4}\right) \]
    limitStart with the original limit.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{12 x^{2} + 12 x + 4}{4 x^{2} + 4 x - 3}\right) \]
    simplify simplifyExpand the squared terms. Distribute the 3 and combine like terms.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{12 + \frac{12}{x} + \frac{4}{x^{2}}}{4 + \frac{4}{x} - \frac{3}{x^{2}}}\right) \]
    algebra simplifyDivide numerator and denominator by x**2. Simplify the fractions inside the limit.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(12 + \frac{12}{x} + \frac{4}{x^{2}}\right) \left(\lim_{x \to \infty}\left(4 + \frac{4}{x} - \frac{3}{x^{2}}\right)\right)^{-1} \]
    limit-lawApply the quotient rule for limits.✓ Proved
  5. \[ = \frac{\lim_{x \to \infty}\left(12 + \frac{12}{x} + \frac{4}{x^{2}}\right)}{4} \]
    limitEvaluate the limit of the denominator.✓ Proved
  6. \[ = 3 \]
    limit simplify simplifyEvaluate the limit of the numerator. Simplify the resulting fraction. Final result.✓ Proved
Answer \( 3 \)
The graph of f(x) = (3(2x + 1)² + 1)/((2x + 1)² - 4). A point runs right along the curve; its height settles toward 3.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{3 \left(2 x + 1\right)^{2} + 1}{\left(2 x + 1\right)^{2} - 4}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{12 x^{2} + 12 x + 4}{4 x^{2} + 4 x - 3}\right) \]
    algebraExpand the squares.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{12 + \frac{12}{x} + \frac{4}{x^{2}}}{4 + \frac{4}{x} - \frac{3}{x^{2}}}\right) \]
    algebraDivide numerator and denominator by x**2.✓ Proved
  4. \[ = 3 \]
    limitAs x→∞, terms with 1/x vanish.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.19

hard Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{4 \left(x + 1\right)^{2} - 1}{x + 2 \left(x + 1\right)^{2} + 1} \).
  1. \[ \lim_{x \to \infty}\left(\frac{4 \left(x + 1\right)^{2} - 1}{x + 2 \left(x + 1\right)^{2} + 1}\right) \]
    limitEvaluate the limit as x approaches infinity.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{4 x^{2} + 8 x + 3}{x + 2 \left(x + 1\right)^{2} + 1}\right) \]
    algebraExpand the squared term in the numerator.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{4 x^{2} + 8 x + 3}{2 x^{2} + 5 x + 3}\right) \]
    algebra algebra simplifyDistribute the 4 in the numerator. Expand the term in the denominator. Combine like terms in the denominator.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{4 + \frac{8}{x} + \frac{3}{x^{2}}}{2 + \frac{5}{x} + \frac{3}{x^{2}}}\right) \]
    algebra simplifyDivide both numerator and denominator by x**2. Simplify the fractions.✓ Proved
  5. \[ = \lim_{x \to \infty} 2 \]
    limit-lawApply the limit laws for the terms approaching zero.✓ Proved
  6. \[ = 2 \]
    limitEvaluate the final value.✓ Proved
Answer \( 2 \)
The graph of f(x) = (4(x + 1)² - 1)/(x + 2(x + 1)² + 1). A point runs right along the curve; its height settles toward 2.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{4 \left(x + 1\right)^{2} - 1}{x + 2 \left(x + 1\right)^{2} + 1}\right) \]
    limit rewriteRewrite denominator to a different but equivalent form.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{4 x^{2} + 8 x + 3}{x + 2 \left(x + 1\right)^{2} + 1}\right) \]
    algebraExpand the numerator.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{4 x^{2} + 8 x + 3}{2 x^{2} + 5 x + 3}\right) \]
    limit-lawExpand the denominator.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{4 + \frac{8}{x} + \frac{3}{x^{2}}}{2 + \frac{5}{x} + \frac{3}{x^{2}}}\right) \]
    rewriteDivide numerator and denominator by x**2.✓ Proved
  5. \[ = 2 \]
    limit✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.379

hard Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{4 \left(3 x + 1\right)^{2} - 1}{3 x + 2 \left(3 x + 1\right)^{2} + 1} \).
  1. \[ \lim_{x \to \infty}\left(\frac{4 \left(3 x + 1\right)^{2} - 1}{3 x + 2 \left(3 x + 1\right)^{2} + 1}\right) \]
    limitStart with the limit of the given function.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{4 \left(3 x + 1\right)^{2} - 1}{3 x + 2 \left(3 x + 1\right)^{2} + 2}\right) \]
    algebraRearrange the denominator terms.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{36 x^{2} + 24 x + 3}{18 x^{2} + 15 x + 4}\right) \]
    simplify algebra simplifyExpand the squared binomial term. Distribute the constants into the parentheses. Combine like terms in the numerator and denominator.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{36 + \frac{24}{x} + \frac{3}{x^{2}}}{18 + \frac{15}{x} + \frac{4}{x^{2}}}\right) \]
    algebra simplifyDivide the numerator and denominator by x**2. Simplify the fractions within the expression.✓ Proved
  5. \[ = \lim_{x \to \infty} 2 \]
    limit-lawApply the limit laws for the quotient of limits.✓ Proved
  6. \[ = 2 \]
    limitEvaluate the final numerical value.✓ Proved
Answer \( 2 \)
The graph of f(x) = (4(3x + 1)² - 1)/(3x + 2(3x + 1)² + 1). A point runs right along the curve; its height settles toward 2.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{4 \left(3 x + 1\right)^{2} - 1}{3 x + 2 \left(3 x + 1\right)^{2} + 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{6 x + 1}{3 x + 1}\right) \]
    factor cancel simplifyFactor numerator and denominator Cancel common factor (2*(3*x+1)+1) Expand 2*(3*x+1)-1✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{6 + \frac{1}{x}}{3 + \frac{1}{x}}\right) \]
    rewriteDivide numerator and denominator by x✓ Proved
  4. \[ = 2 \]
    limitEvaluate the limit as x→∞✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.328

hard Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{- 3 x + \left(3 x - 1\right)^{3} + 1}{2 \left(3 x - 1\right)^{3} + 5} \).
  1. \[ \lim_{x \to \infty}\left(\frac{- 3 x + \left(3 x - 1\right)^{3} + 1}{2 \left(3 x - 1\right)^{3} + 5}\right) \]
    limitStart with the limit of the given function as x approaches infinity.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{27 x^{3} - 27 x^{2} + 6 x}{54 x^{3} - 54 x^{2} + 18 x + 3}\right) \]
    simplifyExpand the binomial terms and simplify the numerator and denominator.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{27 - \frac{27}{x} + \frac{6}{x^{2}}}{54 - \frac{54}{x} + \frac{18}{x^{2}} + \frac{3}{x^{3}}}\right) \]
    algebra simplifyDivide both the numerator and the denominator by the highest power of x, which is x**3. Simplify the fractions within the limit.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(27 - \frac{27}{x} + \frac{6}{x^{2}}\right) \left(\lim_{x \to \infty}\left(54 - \frac{54}{x} + \frac{18}{x^{2}} + \frac{3}{x^{3}}\right)\right)^{-1} \]
    limit-lawApply the quotient rule for limits.✓ Proved
  5. \[ = \frac{1}{2} \]
    limit simplifyEvaluate the limits of the numerator and denominator separately as x goes to infinity. Simplify the resulting fraction.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (-3x + (3x - 1)³ + 1)/(2(3x - 1)³ + 5). A point runs right along the curve; its height settles toward 1/2.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{- 3 x + \left(3 x - 1\right)^{3} + 1}{2 \left(3 x - 1\right)^{3} + 5}\right) \]
    limit rewriteRewrite numerator✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{- 3 x + \left(3 x - 1\right)^{3} + 1}{\left(2 + \frac{5}{\left(3 x - 1\right)^{3}}\right) \left(3 x - 1\right)^{3}}\right) \]
    factorFactor (3*x-1)**3 from denominator✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{1}{2 + \frac{5}{\left(3 x - 1\right)^{3}}} - \frac{1}{\left(2 + \frac{5}{\left(3 x - 1\right)^{3}}\right) \left(3 x - 1\right)^{2}}\right) \]
    limit-lawSplit the fraction✓ Proved
  4. \[ = - \lim_{x \to \infty}\left(\frac{1}{\left(2 + \frac{5}{\left(3 x - 1\right)^{3}}\right) \left(3 x - 1\right)^{2}}\right) + \lim_{x \to \infty} \frac{1}{2 + \frac{5}{\left(3 x - 1\right)^{3}}} \]
    limit-lawSeparate limits✓ Proved
  5. \[ = \lim_{x \to \infty} \frac{1}{2 + \frac{5}{\left(3 x - 1\right)^{3}}} \]
    limit limitSecond term tends to 0 Evaluate remaining limit✓ Proved
  6. \[ = \frac{1}{2} \]
    limit limit5/(3*x-1)^3 → 0 Final value✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.417

hard Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{- 3 x + \left(3 x + 1\right)^{3} - 1}{2 \left(3 x + 1\right)^{3} + 5} \).
  1. \[ \lim_{x \to \infty}\left(\frac{- 3 x + \left(3 x + 1\right)^{3} - 1}{2 \left(3 x + 1\right)^{3} + 5}\right) \]
    limitEvaluate the limit as x approaches infinity.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{27 x^{3} + 27 x^{2} + 6 x}{54 x^{3} + 54 x^{2} + 18 x + 7}\right) \]
    simplify simplifyExpand the cubic term (3x+1)**3. Combine like terms in the numerator and denominator.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{27 + \frac{27}{x} + \frac{6}{x^{2}}}{54 + \frac{54}{x} + \frac{18}{x^{2}} + \frac{7}{x^{3}}}\right) \]
    algebra simplifyDivide both numerator and denominator by the highest power of x, which is x**3. Simplify the fractions by dividing each term by its power of x.✓ Proved
  4. \[ = \lim_{x \to \infty} \frac{1}{2} \]
    limit-lawApply the limit laws as terms with 1/x^n approach zero.✓ Proved
  5. \[ = \frac{1}{2} \]
    simplifyReduce the fraction to its simplest form.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (-3x + (3x + 1)³ - 1)/(2(3x + 1)³ + 5). A point runs right along the curve; its height settles toward 1/2.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{- 3 x + \left(3 x + 1\right)^{3} - 1}{2 \left(3 x + 1\right)^{3} + 5}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{27 x^{3} + 27 x^{2} + 6 x}{54 x^{3} + 54 x^{2} + 18 x + 7}\right) \]
    simplify✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{x \left(27 x^{2} + 27 x + 6\right)}{x \left(54 x^{2} + 54 x + 18\right) + 7}\right) \]
    factorRewrite numerator and denominator with a common factor x.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{27 x^{2} + 27 x + 6}{54 x^{2} + 54 x + 18 + \frac{7}{x}}\right) \]
    simplifyCancel the common factor x.✓ Proved
  5. \[ = \lim_{x \to \infty}\left(\frac{27 + \frac{27}{x} + \frac{6}{x^{2}}}{54 + \frac{54}{x} + \frac{18}{x^{2}} + \frac{7}{x^{3}}}\right) \]
    simplifyDivide numerator and denominator by x**2.✓ Proved
  6. \[ = \lim_{x \to \infty}\left(27 + \frac{27}{x} + \frac{6}{x^{2}}\right) \left(\lim_{x \to \infty}\left(54 + \frac{54}{x} + \frac{18}{x^{2}} + \frac{7}{x^{3}}\right)\right)^{-1} \]
    limit-lawSeparate the limit of a quotient into two limits.✓ Proved
  7. \[ = \frac{1}{2} \]
    limit simplifyEvaluate each limit: numerator → 27, denominator → 54.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.100

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{2 x}{\sqrt{\left(2 x - 1\right)^{2} + 4}} \).
  1. \[ \lim_{x \to \infty}\left(\frac{2 x}{\sqrt{\left(2 x - 1\right)^{2} + 4}}\right) \]
    limitStarting with the original limit expression.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{2 x}{\sqrt{4 x^{2} - 4 x + 5}}\right) \]
    algebra simplifyExpand the squared term inside the square root. Combine the constant terms inside the square root.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{2}{\sqrt{\frac{4 x^{2} - 4 x + 5}{x^{2}}}}\right) \]
    rewrite limit-lawDivide the numerator and denominator by x (noting x > 0 as x approaches infinity). Apply the limit law for the quotient of two functions.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{2}{\sqrt{4 - \frac{4}{x} + \frac{5}{x^{2}}}}\right) \]
    simplifyDistribute the division by x**2 inside the square root.✓ Proved
  5. \[ = \lim_{x \to \infty} 1 \]
    limit-law simplifyEvaluate the limits of the individual terms as x approaches infinity. Simplify the expression inside the square root.✓ Proved
  6. \[ = 1 \]
    simplify simplifyEvaluate the square root and the final division. Final result.✓ Proved
Answer \( 1 \)
The graph of f(x) = 2x/sqrt((2x - 1)² + 4). A point runs right along the curve; its height settles toward 1.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{2 x}{\sqrt{\left(2 x - 1\right)^{2} + 4}}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{2 x}{\sqrt{4 x^{2} - 4 x + 5}}\right) \]
    rewriteExpand the square inside the sqrt.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{2}{\sqrt{4 - \frac{4}{x} + \frac{5}{x^{2}}}}\right) \]
    factor cancelFactor x**2 out of the sqrt. Cancel the common factor x.✓ Proved
  4. \[ = 1 \]
    limit simplifyAs x→∞, the terms with 1/x vanish. Compute the numeric value.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.177

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{3 x + 1}{\sqrt{9 x^{2} + 4}} \).
  1. \[ \lim_{x \to \infty}\left(\frac{3 x + 1}{\sqrt{9 x^{2} + 4}}\right) \]
    limitStart with the original limit expression.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{3 x}{\sqrt{9 x^{2} + 4}} + \frac{1}{\sqrt{9 x^{2} + 4}}\right) \]
    algebraSplit the fraction into two parts.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{3 x}{\sqrt{9 x^{2} + 4}}\right) + \lim_{x \to \infty} \frac{1}{\sqrt{9 x^{2} + 4}} \]
    limit-lawApply the sum rule for limits.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{3 x}{\sqrt{9 x^{2} + 4}}\right) \]
    limitThe second term approaches zero as x goes to infinity.✓ Proved
  5. \[ = \lim_{x \to \infty}\left(\frac{3 x}{\sqrt{9 + \frac{4}{x^{2}}} \left|{x}\right|}\right) \]
    algebraFactor x**2 out of the square root.✓ Proved
  6. \[ = \lim_{x \to \infty}\left(\frac{3}{\sqrt{9 + \frac{4}{x^{2}}}}\right) \]
    simplify algebraSimplify the expression inside the limit. Cancel the common factor of x.✓ Proved
  7. \[ = 1 \]
    limit simplify simplifyEvaluate the limit as x approaches infinity. Simplify the resulting numerical expression. Final result.✓ Proved
Answer \( 1 \)
The graph of f(x) = (3x + 1)/sqrt(9x² + 4). A point runs right along the curve; its height settles toward 1.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{3 x + 1}{\sqrt{9 x^{2} + 4}}\right) \]
    limitWe want the limit as x approaches infinity.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{3 + \frac{1}{x}}{\sqrt{9 + \frac{4}{x^{2}}}}\right) \]
    factorDivide numerator and denominator by x to simplify.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(3 + \frac{1}{x}\right) \left(\lim_{x \to \infty} \sqrt{9 + \frac{4}{x^{2}}}\right)^{-1} \]
    limit-lawApply the quotient rule for limits.✓ Proved
  4. \[ = 1 \]
    limit simplifyEvaluate each limit separately. Simplify the fraction to get the final value.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.138

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{3 \left(x - 1\right)^{2} + 1}{\left(x - 1\right)^{2} - 4} \).
  1. \[ \lim_{x \to \infty}\left(\frac{3 \left(x - 1\right)^{2} + 1}{\left(x - 1\right)^{2} - 4}\right) \]
    limitStart with the limit of the given function.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{3 x^{2} - 6 x + 4}{x^{2} - 2 x - 3}\right) \]
    algebra simplify simplifyExpand the squared terms. Distribute the 3. Combine constant terms in the numerator and denominator.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(3 x^{2} - 6 x + 4\right)}{\frac{d}{d x} \left(x^{2} - 2 x - 3\right)}\right) \]
    lhopitalApply L'Hopital's rule since the limit is of the form infinity/infinity.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{6 x - 6}{2 x - 2}\right) \]
    simplify factorCompute the derivatives. Factor out common terms.✓ Proved
  5. \[ = \lim_{x \to \infty} 3 \]
    simplifyCancel the common factor (x-1).✓ Proved
  6. \[ = 3 \]
    limitEvaluate the final limit.✓ Proved
Answer \( 3 \)
The graph of f(x) = (3(x - 1)² + 1)/((x - 1)² - 4). A point runs right along the curve; its height settles toward 3.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{3 \left(x - 1\right)^{2} + 1}{\left(x - 1\right)^{2} - 4}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{3 + \frac{1}{\left(x - 1\right)^{2}}}{1 - \frac{4}{\left(x - 1\right)^{2}}}\right) \]
    algebraDivide numerator and denominator by (x-1)**2✓ Proved
  3. \[ = 3 \]
    limitAs x→∞, 1/(x-1)**2 → 0, so the expression tends to 3✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.147

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{\sqrt{4 x^{2} + 1}}{2 x} \).
  1. \[ \lim_{x \to \infty}\left(\frac{\sqrt{4 x^{2} + 1}}{2 x}\right) \]
    limitIdentify the limit to evaluate.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{\sqrt{4 + \frac{1}{x^{2}}} \left|{x}\right|}{2 x}\right) \]
    factor algebra simplifyFactor out x**2 from the radicand. Apply the property sqrt(a*b) = sqrt(a)*sqrt(b). Simplify the square root of x**2.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{\sqrt{4 + \frac{1}{x^{2}}}}{2}\right) \]
    simplify cancelSince x approaches infinity, |x| = x. Cancel the common factor of x in the numerator and denominator.✓ Proved
  4. \[ = \frac{\lim_{x \to \infty} 2}{2} \]
    limit-lawApply the limit law for a quotient.✓ Proved
  5. \[ = 1 \]
    simplify simplifyEvaluate the limit of the constant term. Final simplification.✓ Proved
Answer \( 1 \)
The graph of f(x) = sqrt(4x² + 1)/(2x). A point runs right along the curve; its height settles toward 1.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{\sqrt{4 x^{2} + 1}}{2 x}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{\sqrt{1 + \frac{1}{4 x^{2}}} \left|{x}\right|}{x}\right) \]
    factor simplify✓ Proved
  3. \[ = \lim_{x \to \infty} \sqrt{1 + \frac{1}{4 x^{2}}} \]
    limit-lawabs(x)/x → 1 as x → ∞✓ Proved
  4. \[ = 1 \]
    limit✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.157

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{2 x + 2}{\sqrt{\left(2 x + 1\right)^{2} + 4}} \).
  1. \[ \lim_{x \to \infty}\left(\frac{2 x + 2}{\sqrt{\left(2 x + 1\right)^{2} + 4}}\right) \]
    limitStart with the original limit expression.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{2 x + 2}{\sqrt{4 x^{2} + 4 x + 5}}\right) \]
    algebra simplifyExpand the square inside the square root. Combine the constant terms inside the square root.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{2 + \frac{2}{x}}{\sqrt{\frac{4 x^{2} + 4 x + 5}{x^{2}}}}\right) \]
    rewriteDivide numerator and denominator by x (noting that for x > 0, x = sqrt(x**2)).✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{2 + \frac{2}{x}}{\sqrt{4 + \frac{4}{x} + \frac{5}{x^{2}}}}\right) \]
    algebraDistribute the division by x into the terms.✓ Proved
  5. \[ = \lim_{x \to \infty}\left(2 + \frac{2}{x}\right) \left(\lim_{x \to \infty} \sqrt{4 + \frac{4}{x} + \frac{5}{x^{2}}}\right)^{-1} \]
    limit-lawApply the quotient rule for limits.✓ Proved
  6. \[ = 1 \]
    limit simplify simplifyEvaluate the limit of each part as x approaches infinity, noting that terms with 1/x vanish. Simplify the square root. Final evaluation.✓ Proved
Answer \( 1 \)
The graph of f(x) = (2x + 2)/sqrt((2x + 1)² + 4). A point runs right along the curve; its height settles toward 1.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{2 x + 2}{\sqrt{\left(2 x + 1\right)^{2} + 4}}\right) \]
    limit factorFactor out the constant 2 from the numerator.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{2 x + 2}{x \sqrt{\left(2 + \frac{1}{x}\right)^{2} + \frac{4}{x^{2}}}}\right) \]
    rewriteRewrite the square root: sqrt((2*x + 1)**2 + 4) = x*sqrt((2 + 1/x)**2 + 4/x**2).✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{2 + \frac{2}{x}}{\sqrt{\left(2 + \frac{1}{x}\right)^{2} + \frac{4}{x^{2}}}}\right) \]
    cancelCancel the common factor x.✓ Proved
  4. \[ = 1 \]
    limitAs x→∞, 1/x→0, so the expression tends to 2*1/√(2²+0)=1.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.325

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{\sqrt{\left(x - 1\right)^{2} + 1}}{x - 1} \).
  1. \[ \lim_{x \to \infty}\left(\frac{\sqrt{\left(x - 1\right)^{2} + 1}}{x - 1}\right) \]
    limitStart with the original limit expression.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{\sqrt{x^{2} - 2 x + 2}}{x - 1}\right) \]
    algebra simplifyExpand the squared term inside the square root. Combine the constant terms.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{\sqrt{1 - \frac{2}{x} + \frac{2}{x^{2}}} \left|{x}\right|}{x \left(1 - \frac{1}{x}\right)}\right) \]
    rewrite algebra simplifyFactor out x^2 from the numerator and x from the denominator. Distribute the square root over the product in the numerator. Simplify the square root of x squared.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{\sqrt{1 - \frac{2}{x} + \frac{2}{x^{2}}}}{1 - \frac{1}{x}}\right) \]
    simplifySince x approaches infinity, x/x is 1.✓ Proved
  5. \[ = \lim_{x \to \infty} 1 \]
    infinityEvaluate the limits of the individual terms as x approaches infinity.✓ Proved
  6. \[ = 1 \]
    limitCalculate the final value.✓ Proved
Answer \( 1 \)
The graph of f(x) = sqrt((x - 1)² + 1)/(x - 1). A point runs right along the curve; its height settles toward 1.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{\sqrt{\left(x - 1\right)^{2} + 1}}{x - 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{\sqrt{1 + \frac{1}{\left(x - 1\right)^{2}}} \left|{x - 1}\right|}{x - 1}\right) \]
    factorFactor inside the square root.✓ Proved
  3. \[ = \lim_{x \to \infty} \sqrt{1 + \frac{1}{\left(x - 1\right)^{2}}} \]
    simplify cancelsqrt((x - 1)**2) = |x - 1|, and for x → ∞, |x - 1| = x - 1. Cancel (x - 1) in numerator and denominator.✓ Proved
  4. \[ = 1 \]
    limitAs x → ∞, 1/(x - 1)**2 → 0, so sqrt(1 + 0) = 1.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.124

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{x}{\sqrt{\left(x - 1\right)^{2} + 4}} \).
  1. \[ \lim_{x \to \infty}\left(\frac{x}{\sqrt{\left(x - 1\right)^{2} + 4}}\right) \]
    limitStart with the given limit.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{x}{\sqrt{x^{2} - 2 x + 5}}\right) \]
    algebra simplifyExpand the squared term inside the square root. Combine the constant terms.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{x}{\sqrt{1 - \frac{2}{x} + \frac{5}{x^{2}}} \left|{x}\right|}\right) \]
    rewriteFactor out x**2 from the expression inside the square root.✓ Proved
  4. \[ = \lim_{x \to \infty} \frac{1}{\sqrt{1 - \frac{2}{x} + \frac{5}{x^{2}}}} \]
    algebra simplifyTake x out of the square root, noting that x is positive as x approaches infinity. Cancel the common factor of x in the numerator and denominator.✓ Proved
  5. \[ = \lim_{x \to \infty} 1 \]
    limit-lawApply the limit to each term inside the expression.✓ Proved
  6. \[ = 1 \]
    limitEvaluate the final expression.✓ Proved
Answer \( 1 \)
The graph of f(x) = x/sqrt((x - 1)² + 4). A point runs right along the curve; its height settles toward 1.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{x}{\sqrt{\left(x - 1\right)^{2} + 4}}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{x}{\sqrt{x^{2} - 2 x + 5}}\right) \]
    rewriteExpand the square.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{x}{\sqrt{1 - \frac{2}{x} + \frac{5}{x^{2}}} \left|{x}\right|}\right) \]
    factorFactor x**2 out of the square root.✓ Proved
  4. \[ = \lim_{x \to \infty} \frac{1}{\sqrt{1 - \frac{2}{x} + \frac{5}{x^{2}}}} \]
    simplify cancelsqrt(x**2) = x for x > 0. Cancel the x in numerator and denominator.✓ Proved
  5. \[ = \frac{1}{\sqrt{\lim_{x \to \infty}\left(1 - \frac{2}{x} + \frac{5}{x^{2}}\right)}} \]
    limit-lawApply limit to the inner expression.✓ Proved
  6. \[ = 1 \]
    limit simplifyCompute the inner limit: 1 - 0 + 0 = 1. sqrt(1) = 1, so the limit is 1.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.132

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{- x + \left(x + 1\right)^{3} - 1}{2 \left(x + 1\right)^{3} + 5} \).
  1. \[ \lim_{x \to \infty}\left(\frac{- x + \left(x + 1\right)^{3} - 1}{2 \left(x + 1\right)^{3} + 5}\right) \]
    limitEvaluate the limit of the given function as x approaches infinity.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{x^{3} + 3 x^{2} + 2 x}{2 x^{3} + 6 x^{2} + 6 x + 7}\right) \]
    simplifyExpand and simplify the numerator and denominator.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{x^{3}}{2 x^{3} + 6 x^{2} + 6 x + 7} + \frac{3 x^{2}}{2 x^{3} + 6 x^{2} + 6 x + 7} + \frac{2 x}{2 x^{3} + 6 x^{2} + 6 x + 7}\right) \]
    limit-lawSplit the fraction into three separate parts using the limit law for sums.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{2 x}{2 x^{3} + 6 x^{2} + 6 x + 7}\right) + \lim_{x \to \infty}\left(\frac{3 x^{2}}{2 x^{3} + 6 x^{2} + 6 x + 7}\right) + \lim_{x \to \infty}\left(\frac{x^{3}}{2 x^{3} + 6 x^{2} + 6 x + 7}\right) \]
    limit-lawApply the limit law to each term individually.✓ Proved
  5. \[ = \lim_{x \to \infty} \frac{1}{2 + \frac{6}{x} + \frac{6}{x^{2}} + \frac{7}{x^{3}}} + \lim_{x \to \infty}\left(\frac{3}{2 x + 6 + \frac{6}{x} + \frac{7}{x^{2}}}\right) + \lim_{x \to \infty}\left(\frac{2}{2 x^{2} + 6 x + 6 + \frac{7}{x}}\right) \]
    algebraDivide the numerator and denominator of each term by the highest power of x in the denominator.✓ Proved
  6. \[ = \lim_{x \to \infty} \frac{1}{2 + \frac{6}{x} + \frac{6}{x^{2}} + \frac{7}{x^{3}}} \]
    limit-lawEvaluate the limits of the terms that approach zero.✓ Proved
  7. \[ = \lim_{x \to \infty} \frac{1}{2} \]
    infinityEvaluate the limit as terms with 1/x approach zero.✓ Proved
  8. \[ = \frac{1}{2} \]
    limitFinal evaluation of the limit.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (-x + (x + 1)³ - 1)/(2(x + 1)³ + 5). A point runs right along the curve; its height settles toward 1/2.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{- x + \left(x + 1\right)^{3} - 1}{2 \left(x + 1\right)^{3} + 5}\right) \]
    limitOriginal expression.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{x^{3} + 3 x^{2} + 2 x}{2 x^{3} + 6 x^{2} + 6 x + 7}\right) \]
    algebraExpanded numerator and denominator.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{x \left(x + 1\right) \left(x + 2\right)}{2 x^{3} + 6 x^{2} + 6 x + 7}\right) \]
    factorFactored the numerator.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{1 + \frac{3}{x} + \frac{2}{x^{2}}}{2 + \frac{6}{x} + \frac{6}{x^{2}} + \frac{7}{x^{3}}}\right) \]
    algebraDivided numerator and denominator by x**3.✓ Proved
  5. \[ = \frac{1}{2} \]
    limitAs x→∞, terms with 1/x vanish.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.246

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{4 x + 3}{\left(2 x + 1\right)^{2} + 3} \).
  1. \[ \lim_{x \to \infty}\left(\frac{4 x + 3}{\left(2 x + 1\right)^{2} + 3}\right) \]
    limitStart with the original limit problem.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{4 x + 3}{2 x^{2} + 4 x + 4}\right) \]
    simplify simplifyExpand the denominator. Combine constant terms in the denominator.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{\frac{4}{x} + \frac{3}{x^{2}}}{2 + \frac{4}{x} + \frac{4}{x^{2}}}\right) \]
    algebra simplifyDivide numerator and denominator by x**2. Simplify the fractions within the expression.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{4}{x} + \frac{3}{x^{2}}\right) \left(\lim_{x \to \infty}\left(2 + \frac{4}{x} + \frac{4}{x^{2}}\right)\right)^{-1} \]
    limit-lawApply the quotient rule for limits.✓ Proved
  5. \[ = 0 \]
    limit simplifyEvaluate the limits of the numerator and denominator separately. Final result.✓ Proved
Answer \( 0 \)
The graph of f(x) = (4x + 3)/((2x + 1)² + 3). A point runs right along the curve; its height settles toward 0.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{4 x + 3}{\left(2 x + 1\right)^{2} + 3}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{4 x + 3}{4 x^{2} + 4 x + 4}\right) \]
    algebraexpand the square✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{4 x + 3}{4 x^{2} \left(1 + \frac{1}{x} + \frac{1}{x^{2}}\right)}\right) \]
    factorfactor out 4*x**2✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{1 + \frac{3}{4 x}}{x \left(1 + \frac{1}{x} + \frac{1}{x^{2}}\right)}\right) \]
    rewrite cancelfactor 4*x from numerator cancel a factor of 4*x✓ Proved
  5. \[ = \lim_{x \to \infty}\left(\frac{1 + \frac{3}{4 x}}{x + 1 + \frac{1}{x}}\right) \]
    rewritesimplify the denominator✓ Proved
  6. \[ = 0 \]
    limitthe expression behaves like 1/x as x→∞, so the limit is 0✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.366

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{3 \left(x + 1\right)^{2} + 1}{\left(x + 1\right)^{2} - 4} \).
  1. \[ \lim_{x \to \infty}\left(\frac{3 \left(x + 1\right)^{2} + 1}{\left(x + 1\right)^{2} - 4}\right) \]
    limitEvaluate the limit of the function as x approaches infinity.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{3 x^{2} + 6 x + 4}{x^{2} + 2 x - 3}\right) \]
    algebra simplifyExpand the squared terms. Simplify the numerator and denominator.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{3 + \frac{6}{x} + \frac{4}{x^{2}}}{1 + \frac{2}{x} - \frac{3}{x^{2}}}\right) \]
    algebra simplifyDivide both numerator and denominator by x**2. Simplify the fractions.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(3 + \frac{6}{x} + \frac{4}{x^{2}}\right) \left(\lim_{x \to \infty}\left(1 + \frac{2}{x} - \frac{3}{x^{2}}\right)\right)^{-1} \]
    limit-lawApply the quotient rule for limits.✓ Proved
  5. \[ = 3 \]
    limit simplifyEvaluate the limits of the numerator and denominator separately. Final result.✓ Proved
Answer \( 3 \)
The graph of f(x) = (3(x + 1)² + 1)/((x + 1)² - 4). A point runs right along the curve; its height settles toward 3.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{3 \left(x + 1\right)^{2} + 1}{\left(x + 1\right)^{2} - 4}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{3 + \frac{1}{\left(x + 1\right)^{2}}}{1 - \frac{4}{\left(x + 1\right)^{2}}}\right) \]
    algebraDivide numerator and denominator by (x+1)**2✓ Proved
  3. \[ = 3 \]
    limitAs x→∞, 1/(x+1)**2 and 4/(x+1)**2 go to 0✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.374

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{16 x^{2} - 1}{8 x^{2} + 2 x} \).
  1. \[ \lim_{x \to \infty}\left(\frac{16 x^{2} - 1}{8 x^{2} + 2 x}\right) \]
    limitStarting with the given limit.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{16 x^{2}}{8 x^{2} + 2 x} - \frac{1}{8 x^{2} + 2 x}\right) \]
    algebraSplit the fraction into two parts.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{16 x^{2}}{8 x^{2} + 2 x}\right) - \lim_{x \to \infty} \frac{1}{8 x^{2} + 2 x} \]
    limit-lawApply the limit law for subtraction.✓ Proved
  4. \[ = \lim_{x \to \infty}\left(\frac{16}{8 + \frac{2}{x}}\right) - \lim_{x \to \infty} \frac{1}{8 x^{2} + 2 x} \]
    simplifyDivide numerator and denominator by x**2.✓ Proved
  5. \[ = \lim_{x \to \infty} 2 - \lim_{x \to \infty} \frac{1}{8 x^{2} + 2 x} \]
    simplifyEvaluate the limit of the term 2/x.✓ Proved
  6. \[ = 2 - \lim_{x \to \infty} \frac{1}{8 x^{2} + 2 x} \]
    simplifySimplify the first limit term.✓ Proved
  7. \[ = 2 \]
    limit simplifyEvaluate the remaining limit. Final result.✓ Proved
Answer \( 2 \)
The graph of f(x) = (16x² - 1)/(8x² + 2x). A point runs right along the curve; its height settles toward 2.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{16 x^{2} - 1}{8 x^{2} + 2 x}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{16 - \frac{1}{x^{2}}}{8 + \frac{2}{x}}\right) \]
    algebraDivide numerator and denominator by x**2✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{16}{8 + \frac{2}{x}}\right) \]
    algebraCancel the -1/x**2 term✓ Proved
  4. \[ = \lim_{y \to 0^+}\left(\frac{16}{2 y + 8}\right) \]
    substitutionLet y = 1/x, so x -> oo corresponds to y -> 0✓ Proved
  5. \[ = 2 \]
    limit limitEvaluate the limit as y -> 0 Simplify the fraction✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.107

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{15 x - 7}{9 x + 4} \).
  1. \[ \lim_{x \to \infty}\left(\frac{15 x - 7}{9 x + 4}\right) \]
    limitStart with the original limit expression.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{15 - \frac{7}{x}}{9 + \frac{4}{x}}\right) \]
    algebra simplifyDivide both numerator and denominator by x. Simplify the fractions inside the limit.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(15 - \frac{7}{x}\right) \left(\lim_{x \to \infty}\left(9 + \frac{4}{x}\right)\right)^{-1} \]
    limit-lawApply the quotient rule for limits.✓ Proved
  4. \[ = \lim_{x \to \infty} 15 \left(\lim_{x \to \infty} 9\right)^{-1} \]
    limitEvaluate the limits of the individual terms.✓ Proved
  5. \[ = \frac{5}{3} \]
    simplify simplifyPerform the division. Reduce the fraction to lowest terms.✓ Proved
Answer \( \frac{5}{3} \)
The graph of f(x) = (15x - 7)/(9x + 4). A point runs right along the curve; its height settles toward 5/3.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{15 x - 7}{9 x + 4}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{15 - \frac{7}{x}}{9 + \frac{4}{x}}\right) \]
    algebraFactor out x from numerator and denominator✓ Proved
  3. \[ = \frac{5}{3} \]
    limitAs x→∞, 7/x→0 and 4/x→0, so the fraction tends to 15/9 = 5/3✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.115

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{- x + \left(x - 1\right)^{3} + 1}{2 \left(x - 1\right)^{3} + 5} \).
  1. \[ \lim_{x \to \infty}\left(\frac{- x + \left(x - 1\right)^{3} + 1}{2 \left(x - 1\right)^{3} + 5}\right) \]
    limitStart with the limit of the given function.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{x^{3} - 3 x^{2} + 2 x}{2 x^{3} - 6 x^{2} + 6 x + 3}\right) \]
    algebra simplifyExpand the cubic terms. Simplify the numerator and denominator by combining like terms.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{1 - \frac{3}{x} + \frac{2}{x^{2}}}{2 - \frac{6}{x} + \frac{6}{x^{2}} + \frac{3}{x^{3}}}\right) \]
    algebra simplifyDivide both numerator and denominator by the highest power of x, which is x**3. Simplify each term in the expression.✓ Proved
  4. \[ = \lim_{x \to \infty} \frac{1}{2} \]
    limit-lawApply the limit laws for each term as x approaches infinity.✓ Proved
  5. \[ = \frac{1}{2} \]
    limitEvaluate the final numerical value.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (-x + (x - 1)³ + 1)/(2(x - 1)³ + 5). A point runs right along the curve; its height settles toward 1/2.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{- x + \left(x - 1\right)^{3} + 1}{2 \left(x - 1\right)^{3} + 5}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{x^{3} - 3 x^{2} + 2 x}{2 x^{3} - 6 x^{2} + 6 x + 3}\right) \]
    simplifyExpand the binomials to get a rational function.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{1 - \frac{3}{x} + \frac{2}{x^{2}}}{2 - \frac{6}{x} + \frac{6}{x^{2}} + \frac{3}{x^{3}}}\right) \]
    factor cancelFactor out x**3 from numerator and denominator. Cancel the common factor x**3.✓ Proved
  4. \[ = \frac{1}{2} \]
    limitAs x→∞, the terms with 1/x vanish, giving 1/2.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.168

medium Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{15 x + 3}{9 x + 10} \).
  1. \[ \lim_{x \to \infty}\left(\frac{15 x + 3}{9 x + 10}\right) \]
    limitStart with the original limit expression.✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{15 + \frac{3}{x}}{9 + \frac{10}{x}}\right) \]
    algebra cancelFactor out x from the numerator and denominator. Cancel the common factor of x.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(15 + \frac{3}{x}\right) \left(\lim_{x \to \infty}\left(9 + \frac{10}{x}\right)\right)^{-1} \]
    limit-lawApply the quotient rule for limits.✓ Proved
  4. \[ = \lim_{x \to \infty} 15 \left(\lim_{x \to \infty} 9\right)^{-1} \]
    limit-lawEvaluate the limits of the individual terms.✓ Proved
  5. \[ = \frac{5}{3} \]
    limit simplifyEvaluate the resulting fraction. Simplify the fraction to its lowest terms.✓ Proved
Answer \( \frac{5}{3} \)
The graph of f(x) = (15x + 3)/(9x + 10). A point runs right along the curve; its height settles toward 5/3.
A second solution, written separately

By gpt-oss:20b, without seeing the first. SymPy checked every line.

  1. \[ \lim_{x \to \infty}\left(\frac{15 x + 3}{9 x + 10}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \infty}\left(\frac{5 \left(1 + \frac{1}{5 x}\right)}{3 \left(1 + \frac{10}{9 x}\right)}\right) \]
    factorFactor out the leading x from numerator and denominator.✓ Proved
  3. \[ = \lim_{x \to \infty}\left(\frac{\frac{5}{3} + \frac{1}{3 x}}{1 + \frac{10}{9 x}}\right) \]
    simplifyCancel the common factor 15*x/9*x.✓ Proved
  4. \[ = \frac{5}{3} \]
    limit-law limitEvaluate the limits of the small terms as x -> oo. Final value of the limit.✓ Proved

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

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