Consider the rational expression

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

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

. Although the numerator and denominator both equal zero at

x=3

, the limit of

g(x)

x

approaches

3

is finite. (a) Determine

\displaystyle \lim_{x\to 3} g(x)

. Show every algebraic step used to obtain your result. (b) Explain, in complete sentences, why direct substitution of

x=3

into the original expression fails to produce the limit found in part (a). (c) Let

h

be the function defined by

h(x)=g(x)

for

x\neq 3

and

h(3)=\lim_{x\to 3} g(x)

. Describe how the algebraic manipulation carried out in part (a) justifies that

h

is continuous at

x=3