Which statement best explains why exists and is finite?

MATHEMATICS · Single correct · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

Which statement best explains why

\displaystyle \lim_{x\to3}\dfrac{x^{3}-27}{x-3}

exists and is finite? (a) Since both numerator and denominator approach

0

, the limit must be

0

by direct substitution. (b) The limit cannot exist because the original expression is undefined at

x=3

. (c) Because the numerator contains the factor

(x-3)

, the expression can be simplified to a polynomial that is continuous at

x=3

, so the limit is finite. (d) The denominator approaches

0

faster than the numerator, so the quotient grows without bound; therefore the limit is unbounded.