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Limits by factoring practice problems

Direct substitution gives 0/0; factor, cancel the common factor, and substitute again. 63 problems with worked solutions; in 60 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 1.372

hard Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{- 9 x + 2 \left(3 x + 1\right)^{2} - 5}{3 x - 1} \).
  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{- 9 x + 2 \left(3 x + 1\right)^{2} - 5}{3 x - 1}\right) \]
    limit algebraStart with the limit of the given function. Rearrange the terms in the numerator.✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{18 x^{2} + 3 x - 3}{3 x - 1}\right) \]
    simplify simplify simplify factorExpand the squared binomial. Distribute the 2. Combine like terms in the numerator. Factor out the constant 3.✓ Proved
  3. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\left(2 x + 1\right) \left(9 x - 3\right)}{3 x - 1}\right) \]
    factorFactor the quadratic expression in the numerator.✓ Proved
  4. \[ = \lim_{x \to \frac{1}{3}^+}\left(6 x + 3\right) \]
    cancelCancel the common factor (3*x - 1).✓ Proved
  5. \[ = 5 \]
    limit algebra algebra simplifyEvaluate the limit by substituting x = 1/3. Multiply 2 by 1/3. Add the terms in the parentheses. Final simplification.✓ Proved
Answer \( 5 \)
The graph of f(x) = (-9x + 2(3x + 1)² - 5)/(3x - 1). Two points slide along the curve toward x = 1/3, one from each side; their heights close in on 5.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{- 9 x + 2 \left(3 x + 1\right)^{2} - 5}{3 x - 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{18 x^{2} + 3 x - 3}{3 x - 1}\right) \]
    simplify factor factorExpanded the numerator. Factored out a 3 from the numerator. Factored 3 from the denominator.✓ Proved
  3. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{6 x^{2} + x - 1}{x - \frac{1}{3}}\right) \]
    cancelCancelled the common factor 3.✓ Proved
  4. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\left(x + \frac{1}{2}\right) \left(6 x - 2\right)}{x - \frac{1}{3}}\right) \]
    factorFactored the quadratic numerator.✓ Proved
  5. \[ = \lim_{x \to \frac{1}{3}^+}\left(6 x + 3\right) \]
    cancel simplifyCancelled the common factor (x - 1/3). Simplified the expression.✓ Proved
  6. \[ = 5 \]
    limitEvaluated 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 1.92

hard Video
Evaluate \( \displaystyle \lim_{x \to 1} \frac{- 9 x + 2 \left(3 x - 1\right)^{2} + 1}{3 x - 3} \).
  1. \[ \lim_{x \to 1^+}\left(\frac{- 9 x + 2 \left(3 x - 1\right)^{2} + 1}{3 x - 3}\right) \]
    limit factorIdentify the limit to evaluate. Factor the denominator.✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{18 x^{2} - 21 x + 3}{3 x - 3}\right) \]
    simplify simplify simplify factorExpand the squared term. Distribute the constant 2. Combine like terms in the numerator. Factor out the common factor 3 from the numerator.✓ Proved
  3. \[ = \lim_{x \to 1^+}\left(\frac{6 x^{2} - 7 x + 1}{x - 1}\right) \]
    simplifyCancel the common factor 3.✓ Proved
  4. \[ = \lim_{x \to 1^+}\left(6 x - 1\right) \]
    factor cancelFactor the quadratic expression in the numerator. Cancel the common factor (x - 1).✓ Proved
  5. \[ = 5 \]
    limitEvaluate the remaining limit.✓ Proved
Answer \( 5 \)
The graph of f(x) = (-9x + 2(3x - 1)² + 1)/(3x - 3). Two points slide along the curve toward x = 1, one from each side; their heights close in on 5.
A second solution, written separately

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

  1. \[ \lim_{x \to 1^+}\left(\frac{- 9 x + 2 \left(3 x - 1\right)^{2} + 1}{3 x - 3}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{18 x^{2} - 21 x + 3}{3 x - 3}\right) \]
    simplifyExpanded the numerator✓ Proved
  3. \[ = \lim_{x \to 1^+}\left(\frac{\left(x - 1\right) \left(18 x - 3\right)}{3 x - 3}\right) \]
    factorFactored the numerator✓ Proved
  4. \[ = \lim_{x \to 1^+}\left(6 x - 1\right) \]
    cancel simplifyCancelled (x-1) from numerator and denominator Simplified the fraction✓ Proved
  5. \[ = 5 \]
    limitEvaluated 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 1.371

hard Video
Evaluate \( \displaystyle \lim_{x \to 1} \frac{- 2 x + \left(2 x - 1\right)^{2} + 1}{\left(2 x - 1\right)^{2} - 1} \).
  1. \[ \lim_{x \to 1^+}\left(\frac{- 2 x + \left(2 x - 1\right)^{2} + 1}{\left(2 x - 1\right)^{2} - 1}\right) \]
    limitEvaluate the limit of the given function at x = 1.✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(- 2 x + \left(2 x - 1\right)^{2} + 1\right)}{\frac{d}{d x} \left(\left(2 x - 1\right)^{2} - 1\right)}\right) \]
    lhopitalApply L'Hopital's Rule because the limit is in the form 0/0.✓ Proved
  3. \[ = \lim_{x \to 1^+}\left(\frac{8 x - 6}{8 x - 4}\right) \]
    simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the expression by distributing the constants.✓ Proved
  4. \[ = \lim_{x \to 1^+}\left(1 - \frac{2}{8 x - 4}\right) \]
    algebraSplit the fraction into two parts.✓ Proved
  5. \[ = \lim_{x \to 1^+}\left(1 - \frac{1}{4 x - 2}\right) \]
    simplifySimplify the fraction coefficient.✓ Proved
  6. \[ = \lim_{x \to 1^+} \frac{1}{2} \]
    substitution simplifySubstitute x = 1 into the expression. Evaluate the denominator.✓ Proved
  7. \[ = \frac{1}{2} \]
    limitFinal result.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (-2x + (2x - 1)² + 1)/((2x - 1)² - 1). Two points slide along the curve toward x = 1, one from each side; their heights close in on 1/2.
A second solution, written separately

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

  1. \[ \lim_{x \to 1^+}\left(\frac{- 2 x + \left(2 x - 1\right)^{2} + 1}{\left(2 x - 1\right)^{2} - 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{4 x^{2} - 6 x + 2}{4 x^{2} - 4 x}\right) \]
    algebraExpanded numerator and denominator✓ Proved
  3. \[ = \lim_{x \to 1^+}\left(\frac{4 x^{2} - 6 x + 2}{4 x \left(x - 1\right)}\right) \]
    factorFactored numerator and denominator✓ Proved
  4. \[ = \lim_{x \to 1^+}\left(\frac{2 x - 1}{2 x}\right) \]
    cancelCancelled common factor (x-1)✓ Proved
  5. \[ = \frac{1}{2} \]
    limitEvaluated the remaining expression at x=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 1.41

hard Video
Evaluate \( \displaystyle \lim_{x \to 0} \frac{- 2 x + \left(2 x + 1\right)^{2} - 1}{\left(2 x + 1\right)^{2} - 1} \).
  1. \[ \lim_{x \to 0^+}\left(\frac{- 2 x + \left(2 x + 1\right)^{2} - 1}{\left(2 x + 1\right)^{2} - 1}\right) \]
    limitStart with the limit expression.✓ Proved
  2. \[ = \lim_{x \to 0^+}\left(\frac{4 x^{2} + 2 x}{4 x^{2} + 4 x}\right) \]
    algebra simplify simplifyExpand the squared terms. Combine like terms in the numerator and denominator. Combine like terms in the numerator.✓ Proved
  3. \[ = \lim_{x \to 0^+}\left(\frac{4 x + 2}{4 x + 4}\right) \]
    factor cancelFactor out x from both numerator and denominator. Cancel the common factor x.✓ Proved
  4. \[ = \frac{1}{2} \]
    limitEvaluate the limit by substituting x = 0.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (-2x + (2x + 1)² - 1)/((2x + 1)² - 1). Two points slide along the curve toward x = 0, one from each side; their heights close in on 1/2.
A second solution, written separately

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

  1. \[ \lim_{x \to 0^+}\left(\frac{- 2 x + \left(2 x + 1\right)^{2} - 1}{\left(2 x + 1\right)^{2} - 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to 0^+}\left(\frac{4 x^{2} + 2 x}{4 x^{2} + 4 x}\right) \]
    algebraexpand numerator and denominator✓ Proved
  3. \[ = \lim_{x \to 0^+}\left(\frac{2 x + 1}{2 x + 2}\right) \]
    cancelcancel common factor 2*x+1✓ Proved
  4. \[ = \lim_{x \to 0^+}\left(2 x + 1\right) \left(\lim_{x \to 0^+}\left(2 x + 2\right)\right)^{-1} \]
    limit-lawapply limit law for quotient✓ Proved
  5. \[ = \frac{1}{2} \]
    substitution limitevaluate each limit 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 1.140

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{3}{2}} \frac{2 x + \left(2 x - 1\right)^{2} - 7}{2 x - 3} \).
  1. \[ \lim_{x \to \frac{3}{2}^+}\left(\frac{2 x + \left(2 x - 1\right)^{2} - 7}{2 x - 3}\right) \]
    limitStarting with the limit of the function at x = 3/2.✓ Proved
  2. \[ = \lim_{x \to \frac{3}{2}^+}\left(\frac{4 x^{2} - 2 x - 6}{2 x - 3}\right) \]
    algebra simplify factorExpand the squared term. Combine like terms in the numerator. Factor out a 2 from the numerator.✓ Proved
  3. \[ = \lim_{x \to \frac{3}{2}^+}\left(\frac{\left(x + 1\right) \left(4 x - 6\right)}{2 x - 3}\right) \]
    factorFactor the quadratic expression in the numerator.✓ Proved
  4. \[ = \lim_{x \to \frac{3}{2}^+}\left(2 x + 2\right) \]
    cancel simplifyCancel the common factor (2*x - 3). Distribute the 2.✓ Proved
  5. \[ = 5 \]
    substitution algebra limitSubstitute x = 3/2 into the expression. Multiply 2 by 3/2. Final evaluation of the limit.✓ Proved
Answer \( 5 \)
The graph of f(x) = (2x + (2x - 1)² - 7)/(2x - 3). Two points slide along the curve toward x = 3/2, one from each side; their heights close in on 5.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{3}{2}^+}\left(\frac{2 x + \left(2 x - 1\right)^{2} - 7}{2 x - 3}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{3}{2}^+}\left(\frac{\frac{d}{d x} \left(2 x + \left(2 x - 1\right)^{2} - 7\right)}{\frac{d}{d x} \left(2 x - 3\right)}\right) \]
    lhopitalApply l'Hôpital's rule✓ Proved
  3. \[ = \lim_{x \to \frac{3}{2}^+}\left(4 x - 1\right) \]
    simplify simplify simplifyCompute derivatives Expand the numerator Reduce the fraction✓ Proved
  4. \[ = 5 \]
    limitEvaluate the linear function at x = 3/2✓ 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 1.717

medium Video
Evaluate \( \displaystyle \lim_{x \to -1} \frac{\left(2 x - 1\right)^{2} - 9}{2 x + 2} \).
  1. \[ \lim_{x \to -1^+}\left(\frac{\left(2 x - 1\right)^{2} - 9}{2 x + 2}\right) \]
    limitStart with the limit of the function as x approaches -1 from the right.✓ Proved
  2. \[ = \lim_{x \to -1^+}\left(\frac{4 x^{2} - 4 x - 8}{2 x + 2}\right) \]
    simplify simplify factorExpand the squared term in the numerator. Combine constant terms in the numerator. Factor out the common constant 4.✓ Proved
  3. \[ = \lim_{x \to -1^+}\left(\frac{\left(x + 1\right) \left(4 x - 8\right)}{2 x + 2}\right) \]
    factor factorFactor the quadratic expression in the numerator. Factor the denominator.✓ Proved
  4. \[ = \lim_{x \to -1^+}\left(2 x - 4\right) \]
    cancel simplifyCancel the common factor (x + 1). Simplify the remaining fraction.✓ Proved
  5. \[ = -6 \]
    limitEvaluate the limit by substituting x = -1.✓ Proved
Answer \( -6 \)
The graph of f(x) = ((2x - 1)² - 9)/(2x + 2). Two points slide along the curve toward x = -1, one from each side; their heights close in on -6.
A second solution, written separately

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

  1. \[ \lim_{x \to -1^+}\left(\frac{\left(2 x - 1\right)^{2} - 9}{2 x + 2}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to -1^+}\left(\frac{4 x^{2} - 4 x - 8}{2 x + 2}\right) \]
    algebra factorExpand the square and combine constants. Factor out the common 4.✓ Proved
  3. \[ = \lim_{x \to -1^+}\left(\frac{\left(x + 1\right) \left(4 x - 8\right)}{2 x + 2}\right) \]
    factorFactor the quadratic.✓ Proved
  4. \[ = \lim_{x \to -1^+}\left(2 x - 4\right) \]
    cancel simplifyCancel the common (x+1) factor. Simplify the constant factor.✓ Proved
  5. \[ = -6 \]
    limitEvaluate the linear expression at x = -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 1.771

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{\left(3 x + 1\right)^{2} - 4}{3 x - 1} \).
  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\left(3 x + 1\right)^{2} - 4}{3 x - 1}\right) \]
    limit simplifyStart with the limit of the function as x approaches 1/3 from the right. Expand the numerator.✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{9 x^{2} + 6 x - 3}{3 x - 1}\right) \]
    simplify factor factorCombine constant terms in the numerator. Factor out the common factor 3 from the numerator. Factor the quadratic expression in the numerator.✓ Proved
  3. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\left(x + 1\right) \left(9 x - 3\right)}{3 x - 1}\right) \]
    algebraCancel the common factor (3*x - 1) from the numerator and denominator.✓ Proved
  4. \[ = \lim_{x \to \frac{1}{3}^+}\left(3 x + 3\right) \]
    limitEvaluate the limit by substituting x = 1/3.✓ Proved
  5. \[ = 4 \]
    simplify simplifyPerform the addition inside the parentheses. Final result.✓ Proved
Answer \( 4 \)
The graph of f(x) = ((3x + 1)² - 4)/(3x - 1). Two points slide along the curve toward x = 1/3, one from each side; their heights close in on 4.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\left(3 x + 1\right)^{2} - 4}{3 x - 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{9 x^{2} + 6 x - 3}{3 x - 1}\right) \]
    algebra factorExpand the square. Factor out the common 3.✓ Proved
  3. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\left(x + 1\right) \left(9 x - 3\right)}{3 x - 1}\right) \]
    factorFactor the quadratic.✓ Proved
  4. \[ = \lim_{x \to \frac{1}{3}^+}\left(3 x + 3\right) \]
    cancelCancel the common factor (3*x-1).✓ Proved
  5. \[ = 4 \]
    limitEvaluate the simplified expression at x = 1/3.✓ 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 1.811

medium Video
Evaluate \( \displaystyle \lim_{x \to - \frac{2}{3}} \frac{\left(3 x - 1\right)^{2} - 9}{3 x + 2} \).
  1. \[ \lim_{x \to - \frac{2}{3}^+}\left(\frac{\left(3 x - 1\right)^{2} - 9}{3 x + 2}\right) \]
    limit algebraSet up the limit with the specified direction. Ensure the numerator and denominator are clearly grouped.✓ Proved
  2. \[ = \lim_{x \to - \frac{2}{3}^+}\left(\frac{9 x^{2} - 6 x - 8}{3 x + 2}\right) \]
    simplify simplifyExpand the squared binomial. Combine constant terms in the numerator.✓ Proved
  3. \[ = \lim_{x \to - \frac{2}{3}^+}\left(3 x - 4\right) \]
    factor cancelFactor the quadratic expression in the numerator. Cancel the common factor (3*x + 2).✓ Proved
  4. \[ = -6 \]
    limit algebra simplifyEvaluate the limit by direct substitution. Multiply the constant. Final result.✓ Proved
Answer \( -6 \)
The graph of f(x) = ((3x - 1)² - 9)/(3x + 2). Two points slide along the curve toward x = -2/3, one from each side; their heights close in on -6.
A second solution, written separately

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

  1. \[ \lim_{x \to - \frac{2}{3}^+}\left(\frac{\left(3 x - 1\right)^{2} - 9}{3 x + 2}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to - \frac{2}{3}^+}\left(\frac{9 x^{2} - 6 x - 8}{3 x + 2}\right) \]
    algebraexpand numerator✓ Proved
  3. \[ = \lim_{x \to - \frac{2}{3}^+}\left(3 x - 4\right) \]
    factor cancelfactor quadratic cancel common factor✓ Proved
  4. \[ = -6 \]
    limitevaluate the linear expression at x = -2/3✓ 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 1.96

medium Video
Evaluate \( \displaystyle \lim_{x \to 1} \frac{- 3 x + 2 \left(x + 1\right)^{2} - 5}{x - 1} \).
  1. \[ \lim_{x \to 1^+}\left(\frac{- 3 x + 2 \left(x + 1\right)^{2} - 5}{x - 1}\right) \]
    limitStart with the limit of the given function.✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{2 x^{2} + x - 3}{x - 1}\right) \]
    algebra algebra simplify rewriteExpand the squared term. Distribute the 2. Combine like terms in the numerator. Rewrite the numerator to reveal a factor of (x-1).✓ Proved
  3. \[ = \lim_{x \to 1^+}\left(\frac{2 x \left(x - 1\right) + 3 x - 3}{x - 1}\right) \]
    factorFactor out (x - 1) from the numerator.✓ Proved
  4. \[ = \lim_{x \to 1^+}\left(2 x + 3\right) \]
    simplifyCancel the common factor (x - 1).✓ Proved
  5. \[ = 5 \]
    limitEvaluate the limit by substituting x = 1.✓ Proved
Answer \( 5 \)
The graph of f(x) = (-3x + 2(x + 1)² - 5)/(x - 1). Two points slide along the curve toward x = 1, one from each side; their heights close in on 5.
A second solution, written separately

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

  1. \[ \lim_{x \to 1^+}\left(\frac{- 3 x + 2 \left(x + 1\right)^{2} - 5}{x - 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{2 x^{2} + x - 3}{x - 1}\right) \]
    algebraExpanded the numerator✓ Proved
  3. \[ = \lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(2 x^{2} + x - 3\right)}{\frac{d}{d x} \left(x - 1\right)}\right) \]
    lhopitalApplied l'Hôpital's rule✓ Proved
  4. \[ = \lim_{x \to 1^+}\left(4 x + 1\right) \]
    simplifyComputed derivatives✓ Proved
  5. \[ = 5 \]
    limitEvaluated 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 1.149

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{2}{3}} \frac{\left(3 x - 1\right)^{2} - 1}{3 x - 2} \).
  1. \[ \lim_{x \to \frac{2}{3}^+}\left(\frac{\left(3 x - 1\right)^{2} - 1}{3 x - 2}\right) \]
    limit simplifyStart with the limit of the function as x approaches 2/3 from the right. The limit is the same from both sides since the function is continuous near 2/3 except at the point itself.✓ Proved
  2. \[ = \lim_{x \to \frac{2}{3}^+}\left(\frac{9 x^{2} - 6 x}{3 x - 2}\right) \]
    algebra simplifyExpand the squared term in the numerator. Simplify the numerator by subtracting the constants.✓ Proved
  3. \[ = \lim_{x \to \frac{2}{3}^+}\left(3 x\right) \]
    factor cancelFactor the numerator to reveal the common factor in the denominator. Cancel the common factor (3*x - 2) from the numerator and denominator.✓ Proved
  4. \[ = 2 \]
    limit simplifyEvaluate the remaining expression at x = 2/3. Final result.✓ Proved
Answer \( 2 \)
The graph of f(x) = ((3x - 1)² - 1)/(3x - 2). Two points slide along the curve toward x = 2/3, one from each side; their heights close in on 2.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{2}{3}^+}\left(\frac{\left(3 x - 1\right)^{2} - 1}{3 x - 2}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{2}{3}^+}\left(\frac{9 x^{2} - 6 x}{3 x - 2}\right) \]
    algebraexpand the numerator✓ Proved
  3. \[ = \lim_{x \to \frac{2}{3}^+}\left(3 x\right) \]
    factor cancelfactor the numerator cancel the common factor✓ Proved
  4. \[ = 2 \]
    limitevaluate the remaining 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 1.163

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{2 x + \left(2 x + 1\right)^{2} - 5}{2 x - 1} \).
  1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + \left(2 x + 1\right)^{2} - 5}{2 x - 1}\right) \]
    limitEvaluate the limit of the given function at x = 1/2.✓ Proved
  2. \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{4 x^{2} + 6 x - 4}{2 x - 1}\right) \]
    simplify simplify algebraExpand the squared term in the numerator. Combine like terms in the numerator. Split the middle term to prepare for factoring.✓ Proved
  3. \[ = \lim_{x \to \frac{1}{2}^+}\left(2 x + 4\right) \]
    factor cancelFactor the numerator. Cancel the common factor (2*x - 1).✓ Proved
  4. \[ = 5 \]
    limit simplifySubstitute x = 1/2 into the simplified expression. Calculate the final value.✓ Proved
Answer \( 5 \)
The graph of f(x) = (2x + (2x + 1)² - 5)/(2x - 1). Two points slide along the curve toward x = 1/2, one from each side; their heights close in on 5.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + \left(2 x + 1\right)^{2} - 5}{2 x - 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{4 x^{2} + 6 x - 4}{2 x - 1}\right) \]
    algebraExpand the numerator✓ Proved
  3. \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{\left(x + 2\right) \left(4 x - 2\right)}{2 x - 1}\right) \]
    factorFactor the numerator✓ Proved
  4. \[ = \lim_{x \to \frac{1}{2}^+}\left(2 x + 4\right) \]
    cancelCancel the common factor✓ Proved
  5. \[ = 5 \]
    limitEvaluate the remaining 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 1.310

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{27 x^{3} - 1}{9 x^{2} - 1} \).
  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{27 x^{3} - 1}{9 x^{2} - 1}\right) \]
    limitEvaluate the limit of the given function as x approaches 1/3.✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\left(3 x - 1\right) \left(9 x^{2} + 3 x + 1\right)}{9 x^{2} - 1}\right) \]
    factorFactor the numerator using the difference of cubes formula.✓ Proved
  3. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{9 x^{2} + 3 x + 1}{3 x + 1}\right) \]
    factor cancelFactor the denominator using the difference of squares formula. Cancel the common factor (3*x - 1).✓ Proved
  4. \[ = \lim_{x \to \frac{1}{3}^+}\left(9 x^{2} + 3 x + 1\right) \left(\lim_{x \to \frac{1}{3}^+}\left(3 x + 1\right)\right)^{-1} \]
    limit-lawApply the quotient rule for limits.✓ Proved
  5. \[ = \frac{3}{2} \]
    substitution simplify simplifySubstitute x = 1/3 into the expression. Simplify the arithmetic terms. Calculate the final value.✓ Proved
Answer \( \frac{3}{2} \)
The graph of f(x) = (27x³ - 1)/(9x² - 1). Two points slide along the curve toward x = 1/3, one from each side; their heights close in on 3/2.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{27 x^{3} - 1}{9 x^{2} - 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{9 x^{2} + 3 x + 1}{3 x + 1}\right) \]
    factor cancelFactor numerator and denominator Cancel common factor (3*x - 1)✓ Proved
  3. \[ = \frac{3}{2} \]
    limit limitEvaluate the simplified expression at x = 1/3 Result of the evaluation✓ 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 1.326

medium Video
Evaluate \( \displaystyle \lim_{x \to 1} \frac{8 x^{3} - 8}{2 x - 2} \).
  1. \[ \lim_{x \to 1^+}\left(\frac{8 x^{3} - 8}{2 x - 2}\right) \]
    limit factorStart with the limit of the function as x approaches 1 from the right. Factor out the constant 8 from the numerator.✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{\left(8 x - 8\right) \left(x^{2} + x + 1\right)}{2 x - 2}\right) \]
    factor algebraFactor the difference of cubes (x^3 - 1). Factor the denominator.✓ Proved
  3. \[ = \lim_{x \to 1^+}\left(4 x^{2} + 4 x + 4\right) \]
    cancel simplifyCancel the common factor (x - 1). Simplify the remaining fraction.✓ Proved
  4. \[ = \lim_{x \to 1^+} 12 \]
    substitutionEvaluate the expression at x = 1.✓ Proved
  5. \[ = 12 \]
    limitCalculate the final value.✓ Proved
Answer \( 12 \)
The graph of f(x) = (8x³ - 8)/(2x - 2). Two points slide along the curve toward x = 1, one from each side; their heights close in on 12.
A second solution, written separately

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

  1. \[ \lim_{x \to 1^+}\left(\frac{8 x^{3} - 8}{2 x - 2}\right) \]
    limit factorFactor the numerator and denominator✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{\left(8 x - 8\right) \left(x^{2} + x + 1\right)}{2 x - 2}\right) \]
    factorFactor x**3-1 as (x-1)(x**2+x+1)✓ Proved
  3. \[ = \lim_{x \to 1^+}\left(4 x^{2} + 4 x + 4\right) \]
    cancel simplifyCancel the common factor (x-1) Simplify the constant factor✓ Proved
  4. \[ = 12 \]
    limit limitEvaluate the polynomial at x=1 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 1.25

medium Video
Evaluate \( \displaystyle \lim_{x \to 0} \frac{- x + \left(x + 1\right)^{2} - 1}{\left(x + 1\right)^{2} - 1} \).
  1. \[ \lim_{x \to 0^+}\left(\frac{- x + \left(x + 1\right)^{2} - 1}{\left(x + 1\right)^{2} - 1}\right) \]
    limitStart with the limit of the given function.✓ Proved
  2. \[ = \lim_{x \to 0^+}\left(\frac{x^{2} + x}{x^{2} + 2 x}\right) \]
    algebra simplify simplifyExpand the squared terms. Simplify the numerator and denominator. Combine like terms in the numerator.✓ Proved
  3. \[ = \lim_{x \to 0^+}\left(\frac{x + 1}{x + 2}\right) \]
    factor cancelFactor out x from both numerator and denominator. Cancel the common factor x.✓ Proved
  4. \[ = 0.5 \]
    limitEvaluate the limit by substituting x = 0.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (-x + (x + 1)² - 1)/((x + 1)² - 1). Two points slide along the curve toward x = 0, one from each side; their heights close in on 1/2.
A second solution, written separately

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

  1. \[ \lim_{x \to 0^+}\left(\frac{- x + \left(x + 1\right)^{2} - 1}{\left(x + 1\right)^{2} - 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to 0^+}\left(\frac{x^{2} + x}{x^{2} + 2 x}\right) \]
    rewrite simplify simplifyExpand (x + 1)**2 Combine constants Combine like terms✓ Proved
  3. \[ = \lim_{x \to 0^+}\left(\frac{x + 1}{x + 2}\right) \]
    factor cancelFactor x from numerator and denominator Cancel common factor x✓ Proved
  4. \[ = \frac{1}{2} \]
    limitEvaluate the remaining expression at x = 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 1.62

medium Video
Evaluate \( \displaystyle \lim_{x \to 1} \frac{4 x^{2} - 4}{2 x - 2} \).
  1. \[ \lim_{x \to 1^+}\left(\frac{4 x^{2} - 4}{2 x - 2}\right) \]
    limit algebraStart with the limit of the original function. Factor out the constant 4 from the numerator.✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{\left(x + 1\right) \left(4 x - 4\right)}{2 x - 2}\right) \]
    factor rewriteFactor the difference of squares in the numerator. Factor the denominator.✓ Proved
  3. \[ = \lim_{x \to 1^+}\left(2 x + 2\right) \]
    cancel simplifyCancel the common factor (x - 1). Simplify the fraction.✓ Proved
  4. \[ = 4 \]
    limitEvaluate the limit by substituting x = 1.✓ Proved
Answer \( 4 \)
The graph of f(x) = (4x² - 4)/(2x - 2). Two points slide along the curve toward x = 1, one from each side; their heights close in on 4.
A second solution, written separately

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

  1. \[ \lim_{x \to 1^+}\left(\frac{4 x^{2} - 4}{2 x - 2}\right) \]
    limit factorFactor the numerator✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{\left(x + 1\right) \left(4 x - 4\right)}{2 x - 2}\right) \]
    factorFactor the denominator✓ Proved
  3. \[ = \lim_{x \to 1^+}\left(2 x + 2\right) \]
    cancel simplifyCancel the common factor (x - 1) Simplify the constant factor✓ Proved
  4. \[ = 4 \]
    limitEvaluate at x = 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 1.143

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{2}{3}} \frac{9 x^{2} + 3 x - 6}{3 x - 2} \).
  1. \[ \lim_{x \to \frac{2}{3}^+}\left(\frac{9 x^{2} + 3 x - 6}{3 x - 2}\right) \]
    limit factorEvaluate the limit of the given function at x = 2/3. Factor out 3 from the numerator.✓ Proved
  2. \[ = \lim_{x \to \frac{2}{3}^+}\left(\frac{\left(x + 1\right) \left(9 x - 6\right)}{3 x - 2}\right) \]
    factorFactor the quadratic expression 3*x**2 + x - 2.✓ Proved
  3. \[ = \lim_{x \to \frac{2}{3}^+}\left(3 x + 3\right) \]
    cancelCancel the common factor (3*x - 2) from the numerator and denominator.✓ Proved
  4. \[ = 5 \]
    limit simplify simplifySubstitute x = 2/3 into the simplified expression. Combine the terms inside the parentheses. Multiply the remaining terms.✓ Proved
Answer \( 5 \)
The graph of f(x) = (9x² + 3x - 6)/(3x - 2). Two points slide along the curve toward x = 2/3, one from each side; their heights close in on 5.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{2}{3}^+}\left(\frac{9 x^{2} + 3 x - 6}{3 x - 2}\right) \]
    limit factorFactor out a 3 from the numerator.✓ Proved
  2. \[ = \lim_{x \to \frac{2}{3}^+}\left(\frac{\left(x + 1\right) \left(9 x - 6\right)}{3 x - 2}\right) \]
    factorFactor the quadratic 3*x**2 + x - 2.✓ Proved
  3. \[ = \lim_{x \to \frac{2}{3}^+}\left(3 x + 3\right) \]
    cancelCancel the common factor (3*x - 2).✓ Proved
  4. \[ = 5 \]
    limitEvaluate the remaining expression at x = 2/3.✓ 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 1.146

medium Video
Evaluate \( \displaystyle \lim_{x \to 1} \frac{9 x^{2} - 15 x + 6}{3 x - 3} \).
  1. \[ \lim_{x \to 1^+}\left(\frac{9 x^{2} - 15 x + 6}{3 x - 3}\right) \]
    limit factorState the initial limit problem. Factor out 3 from the numerator.✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{\left(x - 1\right) \left(9 x - 6\right)}{3 x - 3}\right) \]
    factor factorFactor the quadratic expression in the numerator. Factor out 3 from the denominator.✓ Proved
  3. \[ = \lim_{x \to 1^+}\left(3 x - 2\right) \]
    algebra cancelCancel the common factor of 3. Cancel the common factor (x - 1).✓ Proved
  4. \[ = 1 \]
    limitEvaluate the limit by substituting x = 1.✓ Proved
Answer \( 1 \)
The graph of f(x) = (9x² - 15x + 6)/(3x - 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. \[ \lim_{x \to 1^+}\left(\frac{9 x^{2} - 15 x + 6}{3 x - 3}\right) \]
    limit factorFactor out 3 from the numerator.✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{\left(x - 1\right) \left(9 x - 6\right)}{3 x - 3}\right) \]
    factorFactor the quadratic 3*x**2 - 5*x + 2.✓ Proved
  3. \[ = \lim_{x \to 1^+}\left(3 x - 2\right) \]
    cancel simplifyCancel the common factor (x - 1). Simplify the fraction.✓ Proved
  4. \[ = 1 \]
    limitEvaluate the linear expression at x = 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 1.324

medium Video
Evaluate \( \displaystyle \lim_{x \to -1} \frac{9 x^{2} - 9}{3 x + 3} \).
  1. \[ \lim_{x \to -1^+}\left(\frac{9 x^{2} - 9}{3 x + 3}\right) \]
    limit factorStart with the limit of the given function. Factor the numerator.✓ Proved
  2. \[ = \lim_{x \to -1^+}\left(\frac{\left(x + 1\right) \left(9 x - 9\right)}{3 x + 3}\right) \]
    factor factorFactor the difference of squares. Factor the denominator.✓ Proved
  3. \[ = \lim_{x \to -1^+}\left(3 x - 3\right) \]
    simplifyCancel the common factor (x + 1).✓ Proved
  4. \[ = -6 \]
    limit simplifyEvaluate the expression at x = -1. Final result.✓ Proved
Answer \( -6 \)
The graph of f(x) = (9x² - 9)/(3x + 3). Two points slide along the curve toward x = -1, one from each side; their heights close in on -6.
A second solution, written separately

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

  1. \[ \lim_{x \to -1^+}\left(\frac{9 x^{2} - 9}{3 x + 3}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to -1^+}\left(\frac{\left(x + 1\right) \left(9 x - 9\right)}{3 x + 3}\right) \]
    factorFactor numerator and denominator✓ Proved
  3. \[ = \lim_{x \to -1^+}\left(3 x - 3\right) \]
    cancelCancel the common factor (x+1)✓ Proved
  4. \[ = -6 \]
    limit limitEvaluate the simplified expression at x = -1 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 1.364

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{9 x^{2} - 3 x}{9 x^{2} - 1} \).
  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{9 x^{2} - 3 x}{9 x^{2} - 1}\right) \]
    limitStart with the limit of the given function as x approaches 1/3 from the right.✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{3 x}{3 x + 1}\right) \]
    factor cancelFactor both the numerator and the denominator. Cancel the common factor (3*x - 1).✓ Proved
  3. \[ = \lim_{x \to \frac{1}{3}^+}\left(3 x\right) \left(\lim_{x \to \frac{1}{3}^+}\left(3 x + 1\right)\right)^{-1} \]
    limit-lawApply the quotient rule for limits.✓ Proved
  4. \[ = \frac{1}{2} \]
    limit-law simplify simplifyEvaluate the limits of the numerator and denominator separately. Simplify the expression. Final result.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (9x² - 3x)/(9x² - 1). Two points slide along the curve toward x = 1/3, one from each side; their heights close in on 1/2.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{9 x^{2} - 3 x}{9 x^{2} - 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{18 x - 3}{18 x}\right) \]
    lhopitalApply l'Hôpital's rule because we have 0/0.✓ Proved
  3. \[ = \frac{1}{2} \]
    limit simplify simplifyEvaluate the simplified expression at x = 1/3. Compute the arithmetic. Reduce 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 1.703

medium Video
Evaluate \( \displaystyle \lim_{x \to 1} \frac{\left(2 x - 1\right)^{3} - 1}{\left(2 x - 1\right)^{2} - 1} \).
  1. \[ \lim_{x \to 1^+}\left(\frac{\left(2 x - 1\right)^{3} - 1}{\left(2 x - 1\right)^{2} - 1}\right) \]
    limitStart with the limit of the original function.✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{\left(2 x - 2\right) \left(2 x + \left(2 x - 1\right)^{2}\right)}{\left(2 x - 1\right)^{2} - 1}\right) \]
    factorFactor the numerator using the difference of cubes formula $a^3 - b^3 = (a-b)(a^2+ab+b^2)$.✓ Proved
  3. \[ = \lim_{x \to 1^+}\left(\frac{2 x + \left(2 x - 1\right)^{2}}{2 x}\right) \]
    factor cancel algebraFactor the denominator using the difference of squares formula $a^2 - b^2 = (a-b)(a+b)$. Cancel the common factor $(2x - 2)$ from the numerator and denominator. Simplify the denominator.✓ Proved
  4. \[ = \lim_{x \to 1^+} \frac{3}{2} \]
    substitutionSubstitute $x = 1$ into the expression.✓ Proved
  5. \[ = \frac{3}{2} \]
    limitEvaluate the final numerical value.✓ Proved
Answer \( \frac{3}{2} \)
The graph of f(x) = ((2x - 1)³ - 1)/((2x - 1)² - 1). Two points slide along the curve toward x = 1, one from each side; their heights close in on 3/2.
A second solution, written separately

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

  1. \[ \lim_{x \to 1^+}\left(\frac{\left(2 x - 1\right)^{3} - 1}{\left(2 x - 1\right)^{2} - 1}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to 1^+}\left(\frac{2 x + \left(2 x - 1\right)^{2}}{2 x}\right) \]
    factor cancel simplifyFactor the differences of powers. Cancel the common factor (2*x-1-1). Simplify the denominator (2*x-1+1) to 2*x.✓ Proved
  3. \[ = \frac{3}{2} \]
    limit limitEvaluate the simplified expression at x = 1. The value of the limit is 3/2.✓ 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

More limits by factoring practice

Evaluate \( \displaystyle \lim_{x \to 1} \frac{x^{2} - 1}{x - 1} \).
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Evaluate \( \displaystyle \lim_{x \to \frac{2}{3}} \frac{- 3 x + \left(3 x - 1\right)^{2} + 1}{\left(3 x - 1\right)^{2} - 1} \).
Problem 1.221hard Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{- 6 x + 2 \left(2 x + 1\right)^{2} - 5}{2 x - 1} \).
Problem 1.230hard Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{3}{2}} \frac{- 6 x + 2 \left(2 x - 1\right)^{2} + 1}{2 x - 3} \).
Problem 1.303hard Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 0} \frac{- 3 x + \left(3 x + 1\right)^{2} - 1}{\left(3 x + 1\right)^{2} - 1} \).
Problem 1.467hard Video✓ Nihil obstat