Let $f(x)=\begin{cases} \dfrac{x^2-4}{x-2},& x<2,\\ ke^{x-2},& x\ge 2, \end{cases}$ …

Mathematics

Problem

Let

f(x)=\begin{cases} \dfrac{x^2-4}{x-2},& x<2,\\ ke^{x-2},& x\ge 2, \end{cases}

where

k

is a real constant. a. Part 1: (a) Determine the value of

k

that makes

f

continuous at

x=2

. b. Part 2: (b) Provide a rigorous

\varepsilon

–

\delta

proof that, for the value of

k

found in part (a),

f

is indeed continuous at

x=2

. c. Part 3: (c) State whether

f

is differentiable at

x=2

for the same value of

k

and justify your answer using appropriate limit definitions.