Mathematics

Problem

Define the function

h

h(x)=\begin{cases} k\dfrac{\sin x}{x},& x\ne 0,\\ k,& x=0, \end{cases}

where

k

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

k

that guarantees continuity of

h

x=0

. Justify your conclusion. b. Part 2: (b) For the value of

k

found in part (a), determine whether

h

is differentiable at

x=0

. If it is differentiable, compute

h'(0)

; if not, explain why differentiability fails. c. Part 3: (c) Discuss how the relationship between the limit in part (a) and the derivative in part (b) illustrates the connection between continuity and differentiability at a point.

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