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Limits by direct substitution practice problems

When the function is continuous at the point, the limit is just the value there. 36 problems with worked solutions; in 35 of them every equation is proved by a computer algebra system.

18 worked problems

Try each one before opening the solution. Every line is labeled with what checked it; 18 of these also have a second solution, written separately and checked the same way. How we verify

Problem 1.172

medium Video
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{2}} \frac{\cos{\left(2 x + 1 \right)}}{2 x + 2} \).
  1. \[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{\cos{\left(2 x + 1 \right)}}{2 x + 2}\right) \]
    limit algebra algebra limitEvaluate the limit of the function as x approaches -1/2. Factor out the 2 in the denominator. Rewrite 2*x + 1 as 2*(x + 1/2). Check if the denominator is zero. Since 2*(-1/2) + 2 = 1, the limit can be evaluated by direct substitution.✓ Proved
  2. \[ = 1 \]
    limit simplify simplify simplifySubstitute x = -1/2 into the expression. Simplify the numerator and denominator. Evaluate cos(0). Final result.✓ Proved
Answer \( 1 \)
The graph of f(x) = cos(2x + 1)/(2x + 2). Two points slide along the curve toward x = -1/2, one from each side; their heights close in on 1.
A second solution, written separately

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

  1. \[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{\cos{\left(2 x + 1 \right)}}{2 x + 2}\right) \]
    limit✓ Proved
  2. \[ = 1 \]
    substitution simplify limitPlug in x = -1/2 Compute the arithmetic inside cos(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 1.135

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} 9 x + \left(3 x + 1\right)^{2} + 2 \).
  1. \[ \lim_{x \to \frac{1}{3}^+}\left(9 x + \left(3 x + 1\right)^{2} + 2\right) \]
    limitStart with the limit definition.✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+}\left(9 x^{2} + 15 x + 3\right) \]
    algebra simplifyExpand the squared term. Combine like terms.✓ Proved
  3. \[ = 9 \]
    limit-law algebra algebra algebraEvaluate the expression at the point x = 1/3. Calculate the square and the product. Simplify the first term. Final summation.✓ Proved
Answer \( 9 \)
The graph of f(x) = 9x + (3x + 1)² + 2. Two points slide along the curve toward x = 1/3, one from each side; their heights close in on 9.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{1}{3}^+}\left(9 x + \left(3 x + 1\right)^{2} + 2\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+} 2 + \lim_{x \to \frac{1}{3}^+}\left(9 x\right) + \lim_{x \to \frac{1}{3}^+} \left(3 x + 1\right)^{2} \]
    limit-lawSplit the sum inside the limit.✓ Proved
  3. \[ = 9 \]
    limit limitEvaluate each simple limit. Add the results.✓ 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 1.226

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{3}{2}} 6 x + \left(2 x - 1\right)^{2} - 4 \).
  1. \[ \lim_{x \to \frac{3}{2}^+}\left(6 x + \left(2 x - 1\right)^{2} - 4\right) \]
    limit algebraEvaluate the limit of the function at x = 3/2. Rearrange the terms.✓ Proved
  2. \[ = \lim_{x \to \frac{3}{2}^+}\left(4 x^{2} + 2 x - 3\right) \]
    simplify simplifyExpand the squared binomial. Combine like terms.✓ Proved
  3. \[ = 9 \]
    limit simplify simplifySubstitute the value x = 3/2 into the expression. Evaluate the exponent. Final calculation.✓ Proved
Answer \( 9 \)
The graph of f(x) = 6x + (2x - 1)² - 4. Two points slide along the curve toward x = 3/2, one from each side; their heights close in on 9.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{3}{2}^+}\left(6 x + \left(2 x - 1\right)^{2} - 4\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{3}{2}^+}\left(4 x^{2} + 2 x - 3\right) \]
    rewrite algebraExpand the square (2*x - 1)**2. Combine like terms.✓ Proved
  3. \[ = 9 \]
    limit limitSubstitute x = 3/2. 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 1.82

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{\cos{\left(2 x - 1 \right)}}{2 x} \).
  1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{\cos{\left(2 x - 1 \right)}}{2 x}\right) \]
    limitStart with the original limit expression.✓ Proved
  2. \[ = \lim_{x \to \frac{1}{2}^+} \cos{\left(2 x - 1 \right)} \left(\lim_{x \to \frac{1}{2}^+}\left(2 x\right)\right)^{-1} \]
    limit-lawApply the quotient rule for limits.✓ Proved
  3. \[ = 1 \]
    limit algebra simplify simplifyEvaluate the limits of the numerator and denominator separately. Simplify the expression inside the numerator and denominator. Evaluate the cosine of zero. Final result.✓ Proved
Answer \( 1 \)
The graph of f(x) = cos(2x - 1)/(2x). Two points slide along the curve toward x = 1/2, one from each side; their heights close in on 1.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{\cos{\left(2 x - 1 \right)}}{2 x}\right) \]
    limitStart with the original limit.✓ Proved
  2. \[ = \lim_{x \to \frac{1}{2}^+} \cos{\left(2 x - 1 \right)} \left(\lim_{x \to \frac{1}{2}^+}\left(2 x\right)\right)^{-1} \]
    limit-lawSeparate numerator and denominator using the limit law.✓ Proved
  3. \[ = 1 \]
    limit simplifyEvaluate each limit: cos(0)=1 and 2*(1/2)=1. Simplify the fraction 1/1 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 1.247

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{2 x + 1}{2 x - 3} \).
  1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 1}{2 x - 3}\right) \]
    limitStart with the original limit expression.✓ Proved
  2. \[ = \lim_{x \to \frac{1}{2}^+}\left(2 x + 1\right) \left(\lim_{x \to \frac{1}{2}^+}\left(2 x - 3\right)\right)^{-1} \]
    limit-lawApply the quotient rule for limits.✓ Proved
  3. \[ = -1 \]
    substitution algebra algebra algebraSubstitute the value x = 1/2 into the expression. Simplify the terms in the numerator and denominator. Perform the addition and subtraction. Divide the numerator by the denominator.✓ Proved
Answer \( -1 \)
The graph of f(x) = (2x + 1)/(2x - 3). Two points slide along the curve toward x = 1/2, one from each side; their heights close in on -1.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 1}{2 x - 3}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{x + \frac{1}{2}}{x - \frac{3}{2}}\right) \]
    rewriteFactor out 2 from numerator and denominator✓ Proved
  3. \[ = -1 \]
    limitDirect substitution gives -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 1.153

medium Video
Evaluate \( \displaystyle \lim_{x \to 1} 3 x + \left(x + 1\right)^{2} + 2 \).
  1. \[ \lim_{x \to 1^+}\left(3 x + \left(x + 1\right)^{2} + 2\right) \]
    limitStart with the limit of the function as x approaches 1.✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(x^{2} + 5 x + 3\right) \]
    algebra simplifyExpand the squared binomial term. Combine like terms.✓ Proved
  3. \[ = 9 \]
    limit simplifyEvaluate the expression at x = 1. Calculate the final value.✓ Proved
Answer \( 9 \)
The graph of f(x) = 3x + (x + 1)² + 2. Two points slide along the curve toward x = 1, one from each side; their heights close in on 9.
A second solution, written separately

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

  1. \[ \lim_{x \to 1^+}\left(3 x + \left(x + 1\right)^{2} + 2\right) \]
    limit✓ Proved
  2. \[ = 9 \]
    limitSubstitute x = 1 into the expression.✓ 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 1.166

medium Video
Evaluate \( \displaystyle \lim_{x \to 3} 3 x + \left(x - 1\right)^{2} - 4 \).
  1. \[ \lim_{x \to 3^+}\left(3 x + \left(x - 1\right)^{2} - 4\right) \]
    limitStart with the limit of the given function.✓ Proved
  2. \[ = \lim_{x \to 3^+}\left(x^{2} + x - 3\right) \]
    algebra simplifyExpand the squared term. Combine like terms.✓ Proved
  3. \[ = 9 \]
    limit simplifyEvaluate the limit by substituting x = 3. Calculate the final value.✓ Proved
Answer \( 9 \)
The graph of f(x) = 3x + (x - 1)² - 4. Two points slide along the curve toward x = 3, one from each side; their heights close in on 9.
A second solution, written separately

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

  1. \[ \lim_{x \to 3^+}\left(3 x + \left(x - 1\right)^{2} - 4\right) \]
    limitStart with the original function.✓ Proved
  2. \[ = \lim_{x \to 3^+}\left(x^{2} + x - 3\right) \]
    algebraSimplify the expression inside the limit.✓ Proved
  3. \[ = 9 \]
    limitEvaluate the simplified polynomial at x=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 1.86

medium Video
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\cos{\left(3 x \right)}}{3 x + 1} \).
  1. \[ \lim_{x \to 0^+}\left(\frac{\cos{\left(3 x \right)}}{3 x + 1}\right) \]
    limitStart with the given limit.✓ Proved
  2. \[ = \lim_{x \to 0^+} 1 \]
    substitution simplifySubstitute x = 0 into the expression. Evaluate the constants in the expression.✓ Proved
  3. \[ = 1 \]
    limitThe limit evaluates to 1.✓ Proved
Answer \( 1 \)
The graph of f(x) = cos(3x)/(3x + 1). Two points slide along the curve toward x = 0, one from each side; their heights close in on 1.
A second solution, written separately

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

  1. \[ \lim_{x \to 0^+}\left(\frac{\cos{\left(3 x \right)}}{3 x + 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to 0^+} \cos{\left(3 x \right)} \left(\lim_{x \to 0^+}\left(3 x + 1\right)\right)^{-1} \]
    limit-lawSeparate numerator and denominator limits✓ Proved
  3. \[ = 1 \]
    limit limitcos(3*x) -> 1, 3*x+1 -> 1 as x->0 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

Problem 1.17

medium Video
Evaluate \( \displaystyle \lim_{x \to 1} 4 x^{2} + 6 x - 1 \).
  1. \[ \lim_{x \to 1^+}\left(4 x^{2} + 6 x - 1\right) \]
    limitIdentify the limit to be evaluated.✓ Proved
  2. \[ = 9 \]
    limitSince the function is a polynomial, it is continuous, so we can evaluate by direct substitution.✓ Proved
Answer \( 9 \)
The graph of f(x) = 4x² + 6x - 1. Two points slide along the curve toward x = 1, one from each side; their heights close in on 9.
A second solution, written separately

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

  1. \[ \lim_{x \to 1^+}\left(4 x^{2} + 6 x - 1\right) \]
    limit✓ Proved
  2. \[ = 9 \]
    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 1.20

easy Video
Evaluate \( \displaystyle \lim_{x \to \frac{3}{2}} \sqrt{2 x + 6} \).
  1. \[ \lim_{x \to \frac{3}{2}^+} \sqrt{2 x + 6} \]
    limitStart with the given limit.✓ Proved
  2. \[ = 3 \]
    limit algebra simplify simplifySince the function is continuous at x = 3/2, we can evaluate by direct substitution. Multiply 2 by 3/2. Add the terms inside the square root. Calculate the square root.✓ Proved
Answer \( 3 \)
The graph of f(x) = sqrt(2x + 6). Two points slide along the curve toward x = 3/2, one from each side; their heights close in on 3.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{3}{2}^+} \sqrt{2 x + 6} \]
    limit✓ Proved
  2. \[ = 3 \]
    limitSubstitute x = 3/2 into the expression.✓ 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 1.118

easy Video
Evaluate \( \displaystyle \lim_{x \to 1} \frac{\cos{\left(x - 1 \right)}}{x} \).
  1. \[ \lim_{x \to 1^+}\left(\frac{\cos{\left(x - 1 \right)}}{x}\right) \]
    limitStart with the original limit expression.✓ Proved
  2. \[ = 1 \]
    limit simplify simplify simplifySince the denominator is non-zero, we can evaluate the limit by direct substitution. Simplify the argument of the cosine function. Evaluate the cosine of zero. Final simplification.✓ Proved
Answer \( 1 \)
The graph of f(x) = cos(x - 1)/x. Two points slide along the curve toward x = 1, one from each side; their heights close in on 1.
A second solution, written separately

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

  1. \[ \lim_{x \to 1^+}\left(\frac{\cos{\left(x - 1 \right)}}{x}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to 1^+} \cos{\left(x - 1 \right)} \left(\lim_{x \to 1^+} x\right)^{-1} \]
    limit-lawSeparate the numerator and denominator limits✓ Proved
  3. \[ = 1 \]
    limit limitEvaluate each limit 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

Problem 1.161

easy Video
Evaluate \( \displaystyle \lim_{x \to \frac{2}{3}} \frac{3 x}{3 x - 4} \).
  1. \[ \lim_{x \to \frac{2}{3}^+}\left(\frac{3 x}{3 x - 4}\right) \]
    limitStart with the given limit.✓ Proved
  2. \[ = \lim_{x \to \frac{2}{3}^+} -1 \]
    substitution simplify simplifySubstitute the value x = 2/3 into the expression. Simplify the numerator and denominator. Perform the division in the denominator.✓ Proved
  3. \[ = -1 \]
    limitEvaluate the final numerical value.✓ Proved
Answer \( -1 \)
The graph of f(x) = 3x/(3x - 4). Two points slide along the curve toward x = 2/3, one from each side; their heights close in on -1.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{2}{3}^+}\left(\frac{3 x}{3 x - 4}\right) \]
    limit✓ Proved
  2. \[ = -1 \]
    limit limitSubstitute the point into the continuous function. Simplify the numeric expression.✓ 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 1.3

easy Video
Evaluate \( \displaystyle \lim_{x \to 4} \sqrt{x + 5} \).
  1. \[ \lim_{x \to 4^+} \sqrt{x + 5} \]
    limitStart with the given limit.✓ Proved
  2. \[ = 3 \]
    limit simplify simplifySince the function is continuous at x = 4, we can evaluate by direct substitution. Simplify the expression inside the square root. Calculate the square root.✓ Proved
Answer \( 3 \)
The graph of f(x) = sqrt(x + 5). Two points slide along the curve toward x = 4, one from each side; their heights close in on 3.
A second solution, written separately

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

  1. \[ \lim_{x \to 4^+} \sqrt{x + 5} \]
    limit✓ Proved
  2. \[ = 3 \]
    limitThe limit equals the value of the continuous function at x = 4.✓ 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 1.5

easy Video
Evaluate \( \displaystyle \lim_{x \to 1} \frac{x + 1}{x - 3} \).
  1. \[ \lim_{x \to 1^+}\left(\frac{x + 1}{x - 3}\right) \]
    limitStart with the limit of the function as x approaches 1.✓ Proved
  2. \[ = -1 \]
    substitution algebraSubstitute the value x = 1 into the expression. Simplify the numerator and denominator.✓ Proved
  3. \[ = -1.0 \]
    limitEvaluate the final numerical value.✓ Proved
Answer \( -1 \)
The graph of f(x) = (x + 1)/(x - 3). Two points slide along the curve toward x = 1, one from each side; their heights close in on -1.
A second solution, written separately

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

  1. \[ \lim_{x \to 1^+}\left(\frac{x + 1}{x - 3}\right) \]
    limit✓ Proved
  2. \[ = -1 \]
    limitThe limit evaluates to -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 1.24

easy Video
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\cos{\left(x \right)}}{x + 1} \).
  1. \[ \lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{x + 1}\right) \]
    limitStart with the original limit expression.✓ Proved
  2. \[ = 1 \]
    limit simplify simplifySince the denominator is non-zero, we can evaluate the limit by direct substitution. Evaluate the numerator and denominator. Final result.✓ Proved
Answer \( 1 \)
The graph of f(x) = cos(x)/(x + 1). Two points slide along the curve toward x = 0, one from each side; their heights close in on 1.
A second solution, written separately

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

  1. \[ \lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{x + 1}\right) \]
    limit✓ Proved
  2. \[ = 1 \]
    limitcos(0)=1 and 0+1=1, so the limit equals 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 1.44

easy Video
Evaluate \( \displaystyle \lim_{x \to 2} \frac{x}{x - 4} \).
  1. \[ \lim_{x \to 2^+}\left(\frac{x}{x - 4}\right) \]
    limitStart with the given limit.✓ Proved
  2. \[ = -1 \]
    limitSubstitute the value x = 2 into the expression.✓ Proved
Answer \( -1 \)
The graph of f(x) = x/(x - 4). Two points slide along the curve toward x = 2, one from each side; their heights close in on -1.
A second solution, written separately

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

  1. \[ \lim_{x \to 2^+}\left(\frac{x}{x - 4}\right) \]
    limit✓ Proved
  2. \[ = -1 \]
    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 1.63

easy Video
Evaluate \( \displaystyle \lim_{x \to 2} x^{2} + 3 x - 1 \).
  1. \[ \lim_{x \to 2^+}\left(x^{2} + 3 x - 1\right) \]
    limitIdentify the limit to be evaluated.✓ Proved
  2. \[ = 9 \]
    limitSince the function is a polynomial, it is continuous, so we can evaluate by direct substitution.✓ Proved
Answer \( 9 \)
The graph of f(x) = x² + 3x - 1. Two points slide along the curve toward x = 2, one from each side; their heights close in on 9.
A second solution, written separately

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

  1. \[ \lim_{x \to 2^+}\left(x^{2} + 3 x - 1\right) \]
    limit✓ Proved
  2. \[ = 9 \]
    limitSubstitute x = 2 into the polynomial.✓ 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 1.102

easy Video
Evaluate \( \displaystyle \lim_{x \to -1} \frac{\cos{\left(x + 1 \right)}}{x + 2} \).
  1. \[ \lim_{x \to -1^+}\left(\frac{\cos{\left(x + 1 \right)}}{x + 2}\right) \]
    limitStart with the given limit.✓ Proved
  2. \[ = 1 \]
    limitEvaluate the expression by direct substitution since the denominator is non-zero.✓ Proved
Answer \( 1 \)
The graph of f(x) = cos(x + 1)/(x + 2). Two points slide along the curve toward x = -1, one from each side; their heights close in on 1.
A second solution, written separately

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

  1. \[ \lim_{x \to -1^+}\left(\frac{\cos{\left(x + 1 \right)}}{x + 2}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to -1^+} \cos{\left(x + 1 \right)} \left(\lim_{x \to -1^+}\left(x + 2\right)\right)^{-1} \]
    limit-lawApply limit law for quotient✓ Proved
  3. \[ = 1 \]
    limit limitEvaluate each 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

More limits by direct substitution practice

Evaluate \( \displaystyle \lim_{x \to \frac{5}{3}} \sqrt{3 x + 4} \).
Problem 1.32easy Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{4}{3}} \sqrt{3 x + 5} \).
Problem 1.36easy Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 2} \sqrt{2 x + 5} \).
Problem 1.49easy Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 3} \sqrt{x + 6} \).
Problem 1.50easy Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 0} \frac{x + 2}{x - 2} \).
Problem 1.89easy Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 1} \sqrt{3 x + 6} \).
Problem 1.108easy Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 5} \sqrt{x + 4} \).
Problem 1.128easy Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 1} \frac{2 x}{2 x - 4} \).
Problem 1.174easy Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{5}{2}} \sqrt{2 x + 4} \).
Problem 1.238easy Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{2}{3}} 9 x^{2} + 9 x - 1 \).
Problem 1.61medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{3}} \frac{\cos{\left(3 x + 1 \right)}}{3 x + 2} \).
Problem 1.73medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 0} \frac{3 x + 2}{3 x - 2} \).
Problem 1.76medium Video✓ Every equation proved
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\cos{\left(2 x \right)}}{2 x + 1} \).
Problem 1.112medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{\cos{\left(3 x - 1 \right)}}{3 x} \).
Problem 1.121medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 0} \frac{2 x + 2}{2 x - 2} \).
Problem 1.130medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} 6 x + \left(2 x + 1\right)^{2} + 2 \).
Problem 1.158medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{3 x + 1}{3 x - 3} \).
Problem 1.228medium VideoNot checked
Evaluate \( \displaystyle \lim_{x \to 1} 9 x + \left(3 x - 1\right)^{2} - 4 \).
Problem 1.233medium Video✓ Nihil obstat