Mathematics

Problem

Let the function

g:\mathbb R\to\mathbb R

be defined by

g(x)=\begin{cases}\displaystyle\frac{x^{2}-9}{x-3}, & x\neq3,\\[4pt]k, & x=3.\end{cases}

Answer the following. a. (i) Evaluate

\displaystyle\lim_{x\to3}\frac{x^{2}-9}{x-3}

. b. (ii) Determine the value of the constant

k

so that

g

is continuous at

x=3

. c. (iii) Write the expression for

g(x)

after re-defining it with this value of

k

so that it becomes continuous on the entire real line.