Define the function by h(x)= cases x^ 2 -4 x-2 , & x 2, ;x -2, [4pt]k, & x=2, [4pt]m,…

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

Problem

Define the function

h:\mathbb R\to\mathbb R

h(x)=\begin{cases}\displaystyle\frac{x^{2}-4}{x-2}, & x\neq2,\;x\neq-2,\\[4pt]k, & x=2,\\[4pt]m, & x=-2.\end{cases}

Answer the following questions. (i) Find

\displaystyle\lim_{x\to2}\frac{x^{2}-4}{x-2}

. (ii) Find

\displaystyle\lim_{x\to-2}\frac{x^{2}-4}{x-2}

. (iii) Determine the values of

k

and

m

that make

h

continuous at both

x=2

and

x=-2

. (iv) State the final continuous form of

h(x)

incorporating the values of

k

and

m

obtained above.