Which of the following best explains why exists?

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Which of the following best explains why

\displaystyle \lim_{x\to 3} \frac{x^{2}-9}{x-3}

exists? (a) Differentiating only the numerator and applying L’Hôpital’s Rule gives

\dfrac{2x}{x-3}

; substituting

x=3

causes division by zero, so the limit does not exist. (b) Because the denominator is

0

x=3

while the numerator is not, the Squeeze Theorem guarantees the limit is finite. (c) Factoring shows

\dfrac{x^{2}-9}{x-3}=(x+3)

for

x\neq3

; since

x+3

is continuous at

3

, the limit exists and equals

6

. (d) After factoring

(x^{2}-9)

(x-3)^{2}

, canceling leaves

x-3

; substituting

x=3

gives

0

, so the limit exists and equals

0