∫Calc Practice
Home›Calculus 1›Limits by rationalizing

Limits by rationalizing practice problems

Multiply by the conjugate to clear a square root that makes the expression 0/0. 45 problems with worked solutions; in 43 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 1.40

hard Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{\sqrt{\left(3 x - 1\right)^{2} + 9} - 3}{\left(3 x - 1\right)^{2}} \).
  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\sqrt{\left(3 x - 1\right)^{2} + 9} - 3}{\left(3 x - 1\right)^{2}}\right) \]
    limit rationalizeSet up the limit with the correct direction. Multiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+} \frac{1}{\sqrt{\left(3 x - 1\right)^{2} + 9} + 3} \]
    algebra simplify cancelExpand the squared term in the numerator. Simplify the numerator. Cancel the common factor (3*x - 1)**2.✓ Proved
  3. \[ = \frac{1}{6} \]
    limit algebra simplify simplifyEvaluate the limit by substituting x = 1/3. Simplify the expression inside the square root. Simplify the square root and the sum. Final result.✓ Proved
Answer \( \frac{1}{6} \)
The graph of f(x) = (sqrt((3x - 1)² + 9) - 3)/(3x - 1)². Two points slide along the curve toward x = 1/3, one from each side; their heights close in on 1/6.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\sqrt{\left(3 x - 1\right)^{2} + 9} - 3}{\left(3 x - 1\right)^{2}}\right) \]
    limit✓ Proved
  2. \[ = \lim_{t \to 0^+}\left(\frac{\sqrt{t^{2} + 9} - 3}{t^{2}}\right) \]
    substitutionLet t = 3*x - 1, so t -> 0 as x -> 1/3.✓ Proved
  3. \[ = \lim_{t \to 0^+}\left(\frac{3 \sqrt{\frac{t^{2}}{9} + 1} - 3}{t^{2}}\right) \]
    algebraFactor 3 out of the square root.✓ Proved
  4. \[ = \lim_{t \to 0^+} \frac{1}{6} \]
    limit-law algebra cancelUse sqrt(1+u) ≈ 1 + u/2 for small u. Simplify the numerator. Cancel t**2 in numerator and denominator.✓ Proved
  5. \[ = \frac{1}{6} \]
    limitThe limit evaluates to 1/6.✓ 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 1.137

hard Video
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{3}} \frac{\sqrt{\left(3 x + 1\right)^{2} + 9} - 3}{\left(3 x + 1\right)^{2}} \).
  1. \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\sqrt{\left(3 x + 1\right)^{2} + 9} - 3}{\left(3 x + 1\right)^{2}}\right) \]
    limit algebraDefine the limit of the function at the point x = -1/3. Multiply the numerator and denominator by the conjugate of the numerator.✓ Proved
  2. \[ = \lim_{x \to - \frac{1}{3}^+} \frac{1}{\sqrt{\left(3 x + 1\right)^{2} + 9} + 3} \]
    algebra simplify cancelExpand the squared term in the numerator. Subtract 9 from 9 in the numerator. Cancel the common factor (3*x + 1)**2 from the numerator and denominator.✓ Proved
  3. \[ = \frac{1}{6} \]
    substitution algebra simplify simplifySubstitute x = -1/3 into the expression. Simplify the term inside the parentheses. Evaluate the square root. Final numerical value.✓ Proved
Answer \( \frac{1}{6} \)
The graph of f(x) = (sqrt((3x + 1)² + 9) - 3)/(3x + 1)². Two points slide along the curve toward x = -1/3, one from each side; their heights close in on 1/6.
A second solution, written separately

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

  1. \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\sqrt{\left(3 x + 1\right)^{2} + 9} - 3}{\left(3 x + 1\right)^{2}}\right) \]
    limit✓ Proved
  2. \[ = \lim_{t \to 0^+}\left(\frac{\sqrt{t^{2} + 9} - 3}{t^{2}}\right) \]
    substitutionLet t = 3*x + 1, so t -> 0 as x -> -1/3.✓ Proved
  3. \[ = \lim_{t \to 0^+} \frac{1}{\sqrt{t^{2} + 9} + 3} \]
    rationalize simplifyMultiply numerator and denominator by the conjugate sqrt(t**2+9)+3. Cancel t**2 in numerator and denominator.✓ Proved
  4. \[ = \frac{1}{6} \]
    limit limitEvaluate the limit by substituting t = 0. Simplify 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 1.798

medium Video
Evaluate \( \displaystyle \lim_{x \to 2} \frac{2 x - 4}{\sqrt{2} \sqrt{x} - 2} \).
  1. \[ \lim_{x \to 2^+}\left(\frac{2 x - 4}{\sqrt{2} \sqrt{x} - 2}\right) \]
    limit factorStart with the original limit. Factor out 2 from the numerator.✓ Proved
  2. \[ = \lim_{x \to 2^+}\left(\frac{\sqrt{2} \left(2 x - 4\right)}{2 \left(\sqrt{x} - \sqrt{2}\right)}\right) \]
    algebraFactor out sqrt(2) from the denominator.✓ Proved
  3. \[ = \lim_{x \to 2^+}\left(\frac{\sqrt{2} \left(\sqrt{x} + \sqrt{2}\right) \left(2 x - 4\right)}{2 \left(x - 2\right)}\right) \]
    rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
  4. \[ = \lim_{x \to 2^+}\left(\frac{\sqrt{2} \left(2 \sqrt{x} + 2 \sqrt{2}\right)}{2}\right) \]
    cancelCancel the common factor (x - 2).✓ Proved
  5. \[ = \lim_{x \to 2^+}\left(\sqrt{2} \left(\sqrt{x} + \sqrt{2}\right)\right) \]
    simplify simplifySimplify the constant coefficient. Simplify 2/sqrt(2) to sqrt(2).✓ Proved
  6. \[ = \lim_{x \to 2^+}\left(\sqrt{2} \sqrt{x} + 2\right) \]
    simplifyDistribute sqrt(2) into the parentheses.✓ Proved
  7. \[ = 4 \]
    limit simplify simplifyEvaluate the limit by substituting x = 2. Simplify sqrt(2)*sqrt(2). Final result.✓ Proved
Answer \( 4 \)
The graph of f(x) = (2x - 4)/(sqrt(2)·sqrt(x) - 2). Two points slide along the curve toward x = 2, 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. \[ \lim_{x \to 2^+}\left(\frac{2 x - 4}{\sqrt{2} \sqrt{x} - 2}\right) \]
    limit factorStart with the original expression Factor 2 from the numerator✓ Proved
  2. \[ = \lim_{x \to 2^+}\left(\frac{\sqrt{2} \left(2 x - 4\right)}{2 \left(\sqrt{x} - \sqrt{2}\right)}\right) \]
    factorFactor sqrt(2) from the denominator✓ Proved
  3. \[ = \lim_{x \to 2^+}\left(\sqrt{2} \left(\sqrt{x} + \sqrt{2}\right)\right) \]
    rewriteMultiply numerator by conjugate✓ Proved
  4. \[ = \lim_{x \to 2^+}\left(\frac{\sqrt{2} \left(2 \sqrt{x} + 2 \sqrt{2}\right)}{2}\right) \]
    cancelCancel (sqrt(x)-sqrt(2))✓ Proved
  5. \[ = 4 \]
    limitEvaluate the remaining expression at x=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 1.35

medium Video
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{3}} \frac{\sqrt{3 x + 5} - 2}{3 x + 1} \).
  1. \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\sqrt{3 x + 5} - 2}{3 x + 1}\right) \]
    limit algebraSet up the limit from the right. Multiply the numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to - \frac{1}{3}^+} \frac{1}{\sqrt{3 x + 5} + 2} \]
    simplify algebra cancelSimplify the numerator using the difference of squares. Group the terms to prepare for cancellation. Cancel the common factor (3*x + 1).✓ Proved
  3. \[ = \lim_{x \to - \frac{1}{3}^+} \frac{1}{4} \]
    limit simplify simplify simplifyEvaluate the limit by substituting x = -1/3. Simplify the expression inside the square root. Calculate the square root. Add the numbers in the denominator.✓ Proved
  4. \[ = \frac{1}{4} \]
    simplifyFinal result.✓ Proved
Answer \( \frac{1}{4} \)
The graph of f(x) = (sqrt(3x + 5) - 2)/(3x + 1). Two points slide along the curve toward x = -1/3, one from each side; their heights close in on 1/4.
A second solution, written separately

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

  1. \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\sqrt{3 x + 5} - 2}{3 x + 1}\right) \]
    limit rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to - \frac{1}{3}^+} \frac{1}{\sqrt{3 x + 5} + 2} \]
    simplify simplify cancelExpand the product in the numerator. Combine like terms in the numerator. Cancel the common factor (3*x + 1).✓ Proved
  3. \[ = \frac{1}{4} \]
    limit limitEvaluate the remaining expression at x = -1/3. 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 1.85

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{8}{3}} \frac{\sqrt{3 x + 1} - 3}{3 x - 8} \).
  1. \[ \lim_{x \to \frac{8}{3}^+}\left(\frac{\sqrt{3 x + 1} - 3}{3 x - 8}\right) \]
    limit algebraStart with the limit as stated in the problem. Multiply the numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to \frac{8}{3}^+} \frac{1}{\sqrt{3 x + 1} + 3} \]
    algebra cancel simplifySimplify the numerator. Cancel the common factor (3*x - 8). Simplify the remaining expression.✓ Proved
  3. \[ = \frac{1}{6} \]
    limit algebra simplify simplify simplifyEvaluate the limit by substituting x = 8/3. Simplify the expression inside the square root. Calculate the square root. Simplify the denominator. Final result.✓ Proved
Answer \( \frac{1}{6} \)
The graph of f(x) = (sqrt(3x + 1) - 3)/(3x - 8). Two points slide along the curve toward x = 8/3, one from each side; their heights close in on 1/6.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{8}{3}^+}\left(\frac{\sqrt{3 x + 1} - 3}{3 x - 8}\right) \]
    limit rewrite rationalizeWe want the limit as x approaches 8/3. Factor 3 from the denominator. Multiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to \frac{8}{3}^+} \frac{1}{\sqrt{3 x + 1} + 3} \]
    simplify simplify cancelExpand the difference of squares. Combine like terms in the numerator. Cancel the common factor (3*x-8).✓ Proved
  3. \[ = \frac{1}{6} \]
    limit simplifyEvaluate the remaining expression at x = 8/3. 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 1.106

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{\sqrt{3 x + 3} - 2}{3 x - 1} \).
  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\sqrt{3 x + 3} - 2}{3 x - 1}\right) \]
    limit algebraSet up the limit with the specified direction. Multiply the numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+} \frac{1}{\sqrt{3 x + 3} + 2} \]
    simplify simplify cancelExpand the numerator. Simplify the numerator. Cancel the common factor (3*x - 1).✓ Proved
  3. \[ = \frac{1}{4} \]
    limit simplify simplify simplify simplifyEvaluate the limit by substituting x = 1/3. Simplify the expression inside the square root. Calculate the square root. Simplify the denominator. Final result.✓ Proved
Answer \( \frac{1}{4} \)
The graph of f(x) = (sqrt(3x + 3) - 2)/(3x - 1). Two points slide along the curve toward x = 1/3, one from each side; their heights close in on 1/4.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\sqrt{3 x + 3} - 2}{3 x - 1}\right) \]
    limit rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to \frac{1}{3}^+} \frac{1}{\sqrt{3 x + 3} + 2} \]
    algebra algebra cancelSimplify the numerator. Rewrite 3*x+3-4 as 3*x-1. Cancel the common factor (3*x-1).✓ Proved
  3. \[ = \frac{1}{4} \]
    limit simplifyEvaluate the remaining expression at x = 1/3. 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 1.142

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{3}{2}} \frac{2 x - 3}{\sqrt{2 x + 1} - 2} \).
  1. \[ \lim_{x \to \frac{3}{2}^+}\left(\frac{2 x - 3}{\sqrt{2 x + 1} - 2}\right) \]
    limit algebraIdentify the limit to evaluate. Multiply the numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to \frac{3}{2}^+}\left(\sqrt{2 x + 1} + 2\right) \]
    algebra simplify cancelExpand the denominator using the difference of squares. Simplify the denominator. Cancel the common factor (2*x - 3).✓ Proved
  3. \[ = 4 \]
    limit algebra simplify simplify simplifyEvaluate the limit by substituting x = 3/2. Simplify the expression inside the square root. Calculate the square root. Evaluate the square root. Final result.✓ Proved
Answer \( 4 \)
The graph of f(x) = (2x - 3)/(sqrt(2x + 1) - 2). Two points slide along the curve toward x = 3/2, 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. \[ \lim_{x \to \frac{3}{2}^+}\left(\frac{2 x - 3}{\sqrt{2 x + 1} - 2}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{3}{2}^+}\left(\sqrt{2 x + 1} + 2\right) \]
    rationalize cancelMultiply numerator and denominator by the conjugate. Cancel the common factor (2*x - 3).✓ Proved
  3. \[ = 4 \]
    limitEvaluate the remaining expression 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 1.160

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{5}{2}} \frac{2 x - 5}{\sqrt{2 x - 1} - 2} \).
  1. \[ \lim_{x \to \frac{5}{2}^+}\left(\frac{2 x - 5}{\sqrt{2 x - 1} - 2}\right) \]
    limit rationalizeStart with the limit of the given function. Multiply the numerator and denominator by the conjugate of the denominator.✓ Proved
  2. \[ = \lim_{x \to \frac{5}{2}^+}\left(\sqrt{2 x - 1} + 2\right) \]
    algebra simplify cancelExpand the denominator using the difference of squares. Simplify the expression in the denominator. Cancel the common factor (2*x - 5) from the numerator and denominator.✓ Proved
  3. \[ = 4 \]
    limit algebra simplify simplify simplifyEvaluate the limit by substituting x = 5/2. Simplify the expression inside the square root. Calculate the value inside the square root. Take the square root. Final addition.✓ Proved
Answer \( 4 \)
The graph of f(x) = (2x - 5)/(sqrt(2x - 1) - 2). Two points slide along the curve toward x = 5/2, 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. \[ \lim_{x \to \frac{5}{2}^+}\left(\frac{2 x - 5}{\sqrt{2 x - 1} - 2}\right) \]
    limit rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to \frac{5}{2}^+}\left(\sqrt{2 x - 1} + 2\right) \]
    simplify simplify cancelExpand the product in the denominator. Combine like terms in the denominator. Cancel the common factor (2*x-5).✓ Proved
  3. \[ = 4 \]
    limitEvaluate the remaining expression at x = 5/2.✓ 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 1.367

medium Video
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{3}} \frac{\sqrt{3 x + 2} - 1}{3 x + 1} \).
  1. \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\sqrt{3 x + 2} - 1}{3 x + 1}\right) \]
    limit algebraEvaluate the limit at the point of the indeterminate form. Multiply the numerator and denominator by the conjugate of the numerator.✓ Proved
  2. \[ = \lim_{x \to - \frac{1}{3}^+} \frac{1}{\sqrt{3 x + 2} + 1} \]
    simplify cancel simplifySimplify the numerator using the difference of squares. Cancel the common factor (3*x + 1) from the numerator and denominator. Simplify the remaining expression.✓ Proved
  3. \[ = \frac{1}{2} \]
    limit simplify simplify simplify simplifySubstitute the value x = -1/3 into the expression. Simplify the expression inside the square root. Evaluate the square root. Simplify the denominator. Final result.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (sqrt(3x + 2) - 1)/(3x + 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. \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\sqrt{3 x + 2} - 1}{3 x + 1}\right) \]
    limit rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to - \frac{1}{3}^+} \frac{1}{\sqrt{3 x + 2} + 1} \]
    simplify cancel simplifyExpand the product in the numerator. Cancel the common factor (3*x + 1). Result after cancellation.✓ Proved
  3. \[ = \frac{1}{2} \]
    limitEvaluate the remaining 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 1.127

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{9}{2}} \frac{\sqrt{2} \sqrt{x} - 3}{2 x - 9} \).
  1. \[ \lim_{x \to \frac{9}{2}^+}\left(\frac{\sqrt{2} \sqrt{x} - 3}{2 x - 9}\right) \]
    limit limit algebra rewriteDefine the limit with the specified direction. Remove the direction specification for standard evaluation. Rewrite the numerator using the property sqrt(a)*sqrt(b) = sqrt(a*b). Multiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to \frac{9}{2}^+} \frac{1}{\sqrt{2} \sqrt{x} + 3} \]
    algebra simplifySimplify the numerator using the difference of squares. Cancel the common factor (2*x - 9).✓ Proved
  3. \[ = \frac{1}{6} \]
    limit simplify simplifyEvaluate the limit by substituting x = 9/2. Simplify the square root. Final calculation.✓ Proved
Answer \( \frac{1}{6} \)
The graph of f(x) = (sqrt(2)·sqrt(x) - 3)/(2x - 9). Two points slide along the curve toward x = 9/2, one from each side; their heights close in on 1/6.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{9}{2}^+}\left(\frac{\sqrt{2} \sqrt{x} - 3}{2 x - 9}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to \frac{9}{2}^+}\left(\frac{\frac{d}{d x} \left(\sqrt{2} \sqrt{x} - 3\right)}{\frac{d}{d x} \left(2 x - 9\right)}\right) \]
    lhopital✓ Proved
  3. \[ = \lim_{x \to \frac{9}{2}^+}\left(\frac{\sqrt{2}}{4 \sqrt{x}}\right) \]
    algebra algebra✓ Proved
  4. \[ = \frac{1}{6} \]
    substitution simplify✓ 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 1.134

medium Video
Evaluate \( \displaystyle \lim_{x \to -1} \frac{\sqrt{\left(x + 1\right)^{2} + 9} - 3}{\left(x + 1\right)^{2}} \).
  1. \[ \lim_{x \to -1^+}\left(\frac{\sqrt{\left(x + 1\right)^{2} + 9} - 3}{\left(x + 1\right)^{2}}\right) \]
    limit rationalizeStart with the given limit. Multiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to -1^+} \frac{1}{\sqrt{\left(x + 1\right)^{2} + 9} + 3} \]
    algebra simplify cancelExpand the numerator. Simplify the numerator. Cancel the common factor (x + 1)**2.✓ Proved
  3. \[ = \lim_{x \to -1^+} \frac{1}{6} \]
    substitutionSubstitute x = -1.✓ Proved
  4. \[ = \frac{1}{6} \]
    limit simplify simplifyEvaluate the limit. Simplify the square root. Final result.✓ Proved
Answer \( \frac{1}{6} \)
The graph of f(x) = (sqrt((x + 1)² + 9) - 3)/(x + 1)². Two points slide along the curve toward x = -1, one from each side; their heights close in on 1/6.
A second solution, written separately

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

  1. \[ \lim_{x \to -1^+}\left(\frac{\sqrt{\left(x + 1\right)^{2} + 9} - 3}{\left(x + 1\right)^{2}}\right) \]
    limit rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to -1^+} \frac{1}{\sqrt{\left(x + 1\right)^{2} + 9} + 3} \]
    simplify simplify cancelExpand the product in the numerator. Combine terms in the numerator. Cancel the common factor (x+1)**2.✓ Proved
  3. \[ = \frac{1}{6} \]
    limit limitEvaluate the remaining expression at x = -1. Simplify 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 1.220

medium Video
Evaluate \( \displaystyle \lim_{x \to 5} \frac{x - 5}{\sqrt{x - 1} - 2} \).
  1. \[ \lim_{x \to 5^+}\left(\frac{x - 5}{\sqrt{x - 1} - 2}\right) \]
    limit algebraStart with the limit of the given function. Multiply the numerator and denominator by the conjugate of the denominator.✓ Proved
  2. \[ = \lim_{x \to 5^+}\left(\sqrt{x - 1} + 2\right) \]
    simplify simplify cancelSimplify the denominator using the difference of squares. Combine the constants in the denominator. Cancel the common factor (x - 5).✓ Proved
  3. \[ = 4 \]
    substitution simplify simplify limitSubstitute x = 5 into the expression. Simplify the expression inside the square root. Calculate the square root. Final result.✓ Proved
Answer \( 4 \)
The graph of f(x) = (x - 5)/(sqrt(x - 1) - 2). Two points slide along the curve toward x = 5, 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. \[ \lim_{x \to 5^+}\left(\frac{x - 5}{\sqrt{x - 1} - 2}\right) \]
    limit rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to 5^+}\left(\sqrt{x - 1} + 2\right) \]
    algebra cancelSimplify the product in the denominator. Cancel the common factor (x - 5).✓ Proved
  3. \[ = 4 \]
    limitEvaluate the simplified expression at x = 5.✓ 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 1.420

medium Video
Evaluate \( \displaystyle \lim_{x \to 1} \frac{\sqrt{\left(x - 1\right)^{2} + 9} - 3}{\left(x - 1\right)^{2}} \).
  1. \[ \lim_{x \to 1^+}\left(\frac{\sqrt{\left(x - 1\right)^{2} + 9} - 3}{\left(x - 1\right)^{2}}\right) \]
    limit rationalizeStart with the limit of the given function. Multiply the numerator and denominator by the conjugate of the numerator.✓ Proved
  2. \[ = \lim_{x \to 1^+} \frac{1}{\sqrt{\left(x - 1\right)^{2} + 9} + 3} \]
    algebra simplify cancelExpand the numerator. Simplify the numerator by subtracting 9. Cancel the common factor (x - 1)**2.✓ Proved
  3. \[ = \frac{1}{6} \]
    limit algebra simplify simplifyEvaluate the limit by substituting x = 1. Simplify the expression inside the square root. Take the square root. Final result.✓ Proved
Answer \( \frac{1}{6} \)
The graph of f(x) = (sqrt((x - 1)² + 9) - 3)/(x - 1)². Two points slide along the curve toward x = 1, one from each side; their heights close in on 1/6.
A second solution, written separately

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

  1. \[ \lim_{x \to 1^+}\left(\frac{\sqrt{\left(x - 1\right)^{2} + 9} - 3}{\left(x - 1\right)^{2}}\right) \]
    limit✓ Proved
  2. \[ = \lim_{x \to 1^+} \frac{1}{\sqrt{\left(x - 1\right)^{2} + 9} + 3} \]
    rationalize cancelMultiply numerator and denominator by the conjugate. Cancel the common factor (x-1)**2.✓ Proved
  3. \[ = \frac{1}{6} \]
    limitEvaluate 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 1.799

medium Video
Evaluate \( \displaystyle \lim_{x \to \frac{10}{3}} \frac{\sqrt{3 x - 1} - 3}{3 x - 10} \).
  1. \[ \lim_{x \to \frac{10}{3}^+}\left(\frac{\sqrt{3 x - 1} - 3}{3 x - 10}\right) \]
    limit algebraEvaluate the limit at the point where the denominator is zero. Multiply the numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to \frac{10}{3}^+} \frac{1}{\sqrt{3 x - 1} + 3} \]
    algebra cancel simplifySimplify the numerator. Cancel the common factor (3*x - 10). Simplify the remaining fraction.✓ Proved
  3. \[ = \frac{1}{6} \]
    substitution simplify simplify simplifySubstitute x = 10/3 into the expression. Simplify the expression inside the square root. Evaluate the square root. Final simplification.✓ Proved
Answer \( \frac{1}{6} \)
The graph of f(x) = (sqrt(3x - 1) - 3)/(3x - 10). Two points slide along the curve toward x = 10/3, one from each side; their heights close in on 1/6.
A second solution, written separately

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

  1. \[ \lim_{x \to \frac{10}{3}^+}\left(\frac{\sqrt{3 x - 1} - 3}{3 x - 10}\right) \]
    limit rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to \frac{10}{3}^+} \frac{1}{\sqrt{3 x - 1} + 3} \]
    algebra cancel simplifyExpand the numerator. Factor 3*x-10 from numerator and denominator. Cancel the common factor.✓ Proved
  3. \[ = \frac{1}{6} \]
    limit algebra simplify simplify limitEvaluate the remaining expression at x = 10/3. Simplify inside the square root. Compute the square root. Add the constants. 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 1.10

medium Video
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sqrt{x + 4} - 2}{x} \).
  1. \[ \lim_{x \to 0^+}\left(\frac{\sqrt{x + 4} - 2}{x}\right) \]
    limit algebraStart with the limit as x approaches 0 from the right. Multiply the numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to 0^+} \frac{1}{\sqrt{x + 4} + 2} \]
    simplify cancel simplifySimplify the numerator. Cancel the common factor of x. Simplify the remaining expression.✓ Proved
  3. \[ = \frac{1}{4} \]
    limit simplify simplifyEvaluate the limit by substituting x = 0. Simplify the square root. Final result.✓ Proved
Answer \( \frac{1}{4} \)
The graph of f(x) = (sqrt(x + 4) - 2)/x. Two points slide along the curve toward x = 0, one from each side; their heights close in on 1/4.
A second solution, written separately

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

  1. \[ \lim_{x \to 0^+}\left(\frac{\sqrt{x + 4} - 2}{x}\right) \]
    limit rationalizeStarting limit Multiply by conjugate✓ Proved
  2. \[ = \lim_{x \to 0^+} \frac{1}{\sqrt{x + 4} + 2} \]
    algebra cancelSimplify numerator Cancel x✓ Proved
  3. \[ = \frac{1}{4} \]
    limit limitSubstitute x=0 Compute 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 1.52

medium Video
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sqrt{x^{2} + 9} - 3}{x^{2}} \).
  1. \[ \lim_{x \to 0^+}\left(\frac{\sqrt{x^{2} + 9} - 3}{x^{2}}\right) \]
    limit rationalizeStart with the initial limit expression. Multiply the numerator and denominator by the conjugate of the numerator.✓ Proved
  2. \[ = \lim_{x \to 0^+} \frac{1}{\sqrt{x^{2} + 9} + 3} \]
    algebra simplify cancelExpand the product in the numerator. Simplify the numerator. Cancel the common x**2 term in the numerator and denominator.✓ Proved
  3. \[ = \frac{1}{6} \]
    limit simplify simplifyEvaluate the limit by substituting x = 0. Simplify the square root. Final simplification.✓ Proved
Answer \( \frac{1}{6} \)
The graph of f(x) = (sqrt(x² + 9) - 3)/x². Two points slide along the curve toward x = 0, one from each side; their heights close in on 1/6.
A second solution, written separately

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

  1. \[ \lim_{x \to 0^+}\left(\frac{\sqrt{x^{2} + 9} - 3}{x^{2}}\right) \]
    limit rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to 0^+} \frac{1}{\sqrt{x^{2} + 9} + 3} \]
    algebra cancelExpand the product in the numerator. Cancel the common factor x**2.✓ Proved
  3. \[ = \frac{1}{6} \]
    limit limitEvaluate the remaining expression at x = 0. Simplify 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 1.72

medium Video
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sqrt{4 x^{2} + 9} - 3}{4 x^{2}} \).
  1. \[ \lim_{x \to 0^+}\left(\frac{\sqrt{4 x^{2} + 9} - 3}{4 x^{2}}\right) \]
    limit rationalizeStart with the limit of the original function. Multiply the numerator and denominator by the conjugate of the numerator.✓ Proved
  2. \[ = \lim_{x \to 0^+} \frac{1}{\sqrt{4 x^{2} + 9} + 3} \]
    algebra simplify cancelExpand the product in the numerator. Simplify the numerator by subtracting 9. Cancel the common factor of 4*x**2 from the numerator and denominator.✓ Proved
  3. \[ = \frac{1}{6} \]
    limit simplify simplifyEvaluate the limit by substituting x = 0. Simplify the square root term. Final result.✓ Proved
Answer \( \frac{1}{6} \)
The graph of f(x) = (sqrt(4x² + 9) - 3)/(4x²). Two points slide along the curve toward x = 0, one from each side; their heights close in on 1/6.
A second solution, written separately

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

  1. \[ \lim_{x \to 0^+}\left(\frac{\sqrt{4 x^{2} + 9} - 3}{4 x^{2}}\right) \]
    limit rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to 0^+} \frac{1}{\sqrt{4 x^{2} + 9} + 3} \]
    simplifyCancel the common factor 4*x**2.✓ Proved
  3. \[ = \frac{1}{6} \]
    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 1.97

medium Video
Evaluate \( \displaystyle \lim_{x \to 3} \frac{x - 3}{\sqrt{x + 1} - 2} \).
  1. \[ \lim_{x \to 3^+}\left(\frac{x - 3}{\sqrt{x + 1} - 2}\right) \]
    limit rationalizeStart with the limit of the given function. Multiply the numerator and denominator by the conjugate of the denominator.✓ Proved
  2. \[ = \lim_{x \to 3^+}\left(\sqrt{x + 1} + 2\right) \]
    algebra simplify simplify cancelExpand the denominator product. Simplify the expression in the denominator. Combine like terms in the denominator. Cancel the common factor (x - 3).✓ Proved
  3. \[ = 4 \]
    limit simplifyEvaluate the limit by substituting x = 3. Final calculation.✓ Proved
Answer \( 4 \)
The graph of f(x) = (x - 3)/(sqrt(x + 1) - 2). Two points slide along the curve toward x = 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. \[ \lim_{x \to 3^+}\left(\frac{x - 3}{\sqrt{x + 1} - 2}\right) \]
    limit rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to 3^+}\left(\sqrt{x + 1} + 2\right) \]
    simplify simplify cancelSimplify the product in the denominator. Combine terms in the denominator. Cancel the common factor (x - 3).✓ Proved
  3. \[ = 4 \]
    limitEvaluate the remaining expression 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 1.119

medium Video
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sqrt{2 x + 1} - 1}{2 x} \).
  1. \[ \lim_{x \to 0^+}\left(\frac{\sqrt{2 x + 1} - 1}{2 x}\right) \]
    limit algebraStart with the limit of the original function. Multiply the numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to 0^+} \frac{1}{\sqrt{2 x + 1} + 1} \]
    algebra simplify cancelSimplify the numerator using the difference of squares. Simplify the numerator. Cancel the common factor of 2*x from the numerator and denominator.✓ Proved
  3. \[ = \frac{1}{2} \]
    limit simplify simplifyEvaluate the limit by substituting x = 0. Simplify the square root. Final simplification.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (sqrt(2x + 1) - 1)/(2x). 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. \[ \lim_{x \to 0^+}\left(\frac{\sqrt{2 x + 1} - 1}{2 x}\right) \]
    limit rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to 0^+} \frac{1}{\sqrt{2 x + 1} + 1} \]
    simplify cancelUse (a-b)(a+b)=a^2-b^2. Cancel the factor 2*x.✓ Proved
  3. \[ = \frac{1}{2} \]
    limit limitEvaluate the remaining expression at 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 1.243

medium Video
Evaluate \( \displaystyle \lim_{x \to 1} \frac{\sqrt{x + 3} - 2}{x - 1} \).
  1. \[ \lim_{x \to 1^+}\left(\frac{\sqrt{x + 3} - 2}{x - 1}\right) \]
    limit algebraState the limit with the specified direction. Multiply the numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to 1^+} \frac{1}{\sqrt{x + 3} + 2} \]
    simplify simplify cancelSimplify the numerator. Combine terms in the numerator. Cancel the common factor (x - 1).✓ Proved
  3. \[ = \lim_{x \to 1^+} \frac{1}{4} \]
    limitEvaluate the limit by substituting x = 1.✓ Proved
  4. \[ = \frac{1}{4} \]
    simplify simplifySimplify the denominator. Final result.✓ Proved
Answer \( \frac{1}{4} \)
The graph of f(x) = (sqrt(x + 3) - 2)/(x - 1). Two points slide along the curve toward x = 1, one from each side; their heights close in on 1/4.
A second solution, written separately

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

  1. \[ \lim_{x \to 1^+}\left(\frac{\sqrt{x + 3} - 2}{x - 1}\right) \]
    limit rationalizeMultiply numerator and denominator by the conjugate.✓ Proved
  2. \[ = \lim_{x \to 1^+} \frac{1}{\sqrt{x + 3} + 2} \]
    algebra algebra cancelSimplify the numerator. Rewrite x+3-4 as x-1. Cancel the common factor (x-1).✓ Proved
  3. \[ = \frac{1}{4} \]
    limit limitEvaluate the remaining 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

More limits by rationalizing practice

Evaluate \( \displaystyle \lim_{x \to 4} \frac{x - 4}{\sqrt{x} - 2} \).
Problem 1.14medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sqrt{x + 1} - 1}{x} \).
Problem 1.23medium Video✓ Every equation proved
Evaluate \( \displaystyle \lim_{x \to 3} \frac{\sqrt{3} \sqrt{x} - 3}{3 x - 9} \).
Problem 1.29medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sqrt{3 x + 4} - 2}{3 x} \).
Problem 1.53medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{2}} \frac{\sqrt{2 x + 5} - 2}{2 x + 1} \).
Problem 1.68medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sqrt{9 x^{2} + 9} - 3}{9 x^{2}} \).
Problem 1.75medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{2}} \frac{\sqrt{2 x + 2} - 1}{2 x + 1} \).
Problem 1.78medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 4} \frac{\sqrt{2 x + 1} - 3}{2 x - 8} \).
Problem 1.93medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sqrt{3 x + 1} - 1}{3 x} \).
Problem 1.144medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{\sqrt{2 x + 3} - 2}{2 x - 1} \).
Problem 1.159medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{5}{3}} \frac{3 x - 5}{\sqrt{3 x - 1} - 2} \).
Problem 1.173medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{\sqrt{2} \sqrt{x} - 1}{2 x - 1} \).
Problem 1.224medium VideoReviewed
Evaluate \( \displaystyle \lim_{x \to 1} \frac{\sqrt{x} - 1}{x - 1} \).
Problem 1.236medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 10} \frac{\sqrt{x - 1} - 3}{x - 10} \).
Problem 1.252medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{4}{3}} \frac{3 x - 4}{\sqrt{3} \sqrt{x} - 2} \).
Problem 1.301medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9} \).
Problem 1.354medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 1} \frac{3 x - 3}{\sqrt{3 x + 1} - 2} \).
Problem 1.358medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{\sqrt{3} \sqrt{x} - 1}{3 x - 1} \).
Problem 1.359medium VideoReviewed
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sqrt{2 x + 4} - 2}{2 x} \).
Problem 1.421medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to -1} \frac{\sqrt{x + 5} - 2}{x + 1} \).
Problem 1.463medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to -1} \frac{\sqrt{x + 2} - 1}{x + 1} \).
Problem 1.523medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 5} \frac{\sqrt{2 x - 1} - 3}{2 x - 10} \).
Problem 1.557medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to 8} \frac{\sqrt{x + 1} - 3}{x - 8} \).
Problem 1.714medium Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{\sqrt{\left(2 x - 1\right)^{2} + 9} - 3}{\left(2 x - 1\right)^{2}} \).
Problem 1.116hard Video✓ Nihil obstat
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{2}} \frac{\sqrt{\left(2 x + 1\right)^{2} + 9} - 3}{\left(2 x + 1\right)^{2}} \).
Problem 1.178hard Video✓ Nihil obstat