When evaluating

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

Problem

When evaluating

\displaystyle \lim_{h\to 0}\dfrac{\sqrt{25+h}-5}{h}

, which of the following best explains why multiplying the numerator and denominator by

\sqrt{25+h}+5

is an effective first step? (a) It rationalizes the denominator by removing the radical in the denominator, allowing direct substitution of

h=0

.},{ (b) It distributes the square root, rewriting

\sqrt{25+h}

\sqrt{25}+\sqrt{h}

so that each term can be evaluated separately. (c) It forms a difference of squares

\bigl(\sqrt{25+h}-5\bigr)\bigl(\sqrt{25+h}+5\bigr)=h

, producing a factor that cancels with the denominator. (d) It turns the numerator into

(25+h)^2-25^2

, yielding

h^2

in the numerator and causing the limit to go to

0

.}],