Which of the following statements best explains why ?

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

Problem

Which of the following statements best explains why

\displaystyle \lim_{x\to4}\frac{x^{2}-16}{x-4}=8

? (a) Since

(x^{2}-16)=(x-4)^{2}

, cancelling one factor leaves

(x-4)

, which approaches

0

, so the limit is

0

. (b) Because the factor

x-4

is common to both numerator and denominator, cancelling it gives

x+4

, a polynomial that is continuous at

x=4

, so the limit is

8

. (c) Because

\dfrac{x^{2}-16}{x-4}

is defined at

x=4

, direct substitution gives

8

. (d) Because a

0/0

limit can be evaluated by differentiating only the numerator;

(x^{2}-16)'\big|_{x=4}=8

, so the limit is

8