A rational function is defined by f(x)= x^2-9 x-3 for all

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

Problem

A rational function is defined by

f(x)=\frac{x^2-9}{x-3}

for all

x\ne3

. This problem explores continuity and differentiability at the removable discontinuity. (a) Determine the limit

\displaystyle \lim_{x\to3}f(x)

and state whether

f

can be redefined at

x=3

to make the resulting function

g

continuous on

\mathbb R

. Clearly state the value of

g(3)

if such a redefinition is possible. (b) Using your redefined function

g

, examine whether

g'(3)

exists. Provide a justification that references appropriate limit definitions and algebraic properties of

g

. (c) Explain in words how the existence (or non-existence) of

g'(3)

relates to the manner in which the removable discontinuity was resolved.