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Limits of Rational Functions

A public limits question that resolves a $0/0$ indeterminate form by factoring and canceling a common factor before substituting.

Question

Evaluate lim⁡x→3x2−9x2−5x+6\displaystyle\lim_{x \to 3} \frac{x^2 - 9}{x^2 - 5x + 6}.

  1. a.00
  2. b.11
  3. c.66
  4. d.The limit does not exist

Correct answer

C. 66

Full reasoning

  1. 1Direct substitution gives 9−99−15+6=00\dfrac{9 - 9}{9 - 15 + 6} = \dfrac{0}{0}, an indeterminate form, so the limit laws cannot be applied directly.
  2. 2Factor both polynomials: x2−9=(x−3)(x+3)x^2 - 9 = (x-3)(x+3) and x2−5x+6=(x−3)(x−2)x^2 - 5x + 6 = (x-3)(x-2).
  3. 3Cancel the common factor (x−3)(x-3), valid for x≠3x \neq 3, leaving x+3x−2\dfrac{x+3}{x-2}.
  4. 4Substitute x=3x = 3 into the simplified expression: 3+33−2=6\dfrac{3+3}{3-2} = 6.

Why each choice is right or wrong

Choice A

This substitutes x=3x = 3 into the numerator alone and concludes the limit is zero, ignoring that the denominator also vanishes there.

Choice B

This takes the ratio of the leading coefficients, which is the behavior as x→∞x \to \infty, not the limit at the finite point x=3x = 3.

Choice C

Correct. Factoring gives (x−3)(x+3)(x−3)(x−2)=x+3x−2\dfrac{(x-3)(x+3)}{(x-3)(x-2)} = \dfrac{x+3}{x-2} for x≠3x \neq 3, and substituting yields 61=6\dfrac{6}{1} = 6.

Choice D

This stops at the indeterminate form 0/00/0, which signals more work is needed rather than proving the limit fails to exist.

Related formula

Limit after canceling a common factor

lim⁡x→af(x)g(x)=lim⁡x→aF(x)G(x),f=(x−a)F,  g=(x−a)G\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{F(x)}{G(x)}, \quad f = (x-a)F, \; g = (x-a)G

Assumptions: The simplified denominator G is nonzero at a, so the reduced expression can be evaluated by substitution.

Related topic and practice

Limits and ContinuityReal Analysis Domain GuideCreate a free account to practice

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Sources

Authored for this public explainer registry. It has no database question ID and is not copied from a protected or official exam bank.

  1. Calculus Volume 1, Section 2.3: The Limit Laws — OpenStax. Accessed 2026-08-03. OpenStax textbook content is CC BY-NC-SA 4.0; attribute and avoid verbatim reuse beyond short cited references.
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Sources last checked 2026-08-07

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