Mathematics

Problem

Let

g(x)=\begin{cases} \dfrac{\sin x}{x}, & x\ne0,\\ 1, & x=0. \end{cases}

(a) Verify that

g

is continuous at

x=0

. (b) Evaluate

\displaystyle\lim_{h\to0}\dfrac{g(h)-g(0)}{h}

and decide whether

g

possesses a derivative at

x=0

. (c) For

x\ne0

, derive an explicit expression for

g'(x)

by legitimate algebraic manipulation, and state its domain of validity. a. Part (a) — Continuity verification at the origin b. Part (b) — Limit of the difference quotient at the origin c. Part (c) — Derivative formula for

x\ne0

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