Mathematics

Problem

Consider the function

P(x)=\begin{cases} \dfrac{e^{kx}-1}{x}, & x\neq0,\\ k, & x=0, \end{cases}

where

k

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

k

that guarantees

P

is continuous at

x=0

. Give exact value(s). b. Part 2: (b) For the value of

k

obtained in part (a), show that

P

is differentiable at

x=0

by computing

P'(0)

directly from the definition of the derivative. c. Part 3: (c) Using your result from part (b), write the equation of the tangent line to

y=P(x)

x=0

. Interpret the meaning of the tangent line’s slope in terms of the parameter

k