Mathematics

Problem

Consider the function

h(x)=\begin{cases} \dfrac{\sin(x)-\sin(a)}{x-a}, & x\ne a,\\ b, & x=a, \end{cases}

where

a

and

b

are real numbers with

-\pi<a<\pi

. a. Part 1: (a) Determine the value of

b

in terms of

a

that makes

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

exist. b. Part 2: (b) For that same choice of

b

, decide whether

h

is differentiable at

x=a

. Provide a complete justification based on appropriate limit definitions. c. Part 3: (c) Show that there is no nonzero constant

k

such that replacing

b

with

b+k(x-a)

in the definition of

h

would render the function differentiable of class

C^1

x=a

. d. Part 4: (d) Discuss how your results connect to known limits involving trigonometric difference quotients.