Let g(x)= x^ 3 -8 x-2 , ;x 2

MATHEMATICS · Subjective · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

g(x)=\frac{x^{3}-8}{x-2},\;x\ne 2.

(i) Obtain the simplified expression for

g(x)

by carrying out the division of the numerator by the denominator. (ii) Evaluate

\displaystyle \lim_{x\to 2}g(x)

. Hence define a new function

h(x)=\begin{cases}g(x), & x\ne 2,\\c, & x=2,\end{cases}

where the constant

c

is chosen so that

h(x)

is continuous at

x=2

. (iii) Is

g(x)

a polynomial on its domain? Is the extended function

h(x)

a polynomial on

\mathbb R

? Justify each answer clearly.