Mathematics

Problem

For a real constant

a

, define the function

r_a(x)=\begin{cases} \dfrac{x^{2}-a^{2}}{x-a}, & x\ne a,\\ k, & x=a,\end{cases}

where

k

is a real number to be chosen. a. Part 1: (a) Determine

\displaystyle \lim_{x\to a} r_a(x)

in terms of

a

. b. Part 2: (b) Find the value of

k

that makes

r_a

continuous at

x=a

. Show all steps clearly. c. Part 3: (c) Prove that, with the value of

k

found in part (b), the function

r_a

is continuous for all real numbers

x

. Your argument should address the continuity on

(-\infty,a)

, at

a

, and on

(a,\infty)

separately.