Let p(x) = cases (2x), & x 4

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

p(x) = \begin{cases} \sin(2x), & x \le \frac{\pi}{4},\\ ce^{x-\pi/4}, & x > \frac{\pi}{4}, \end{cases}

where

c

is a positive constant. (a) Compute

\displaystyle \lim_{x\to (\pi/4)^-} p(x)

and

\displaystyle \lim_{x\to (\pi/4)^+} p(x)

in terms of

c

. Show detailed algebraic or trigonometric steps. (b) Determine the value of

c

that makes

p

continuous at

x=\frac{\pi}{4}

. Explain each step in your reasoning. (c) For the value of

c

found in part (b), evaluate

\displaystyle \lim_{x\to (\pi/4)} \dfrac{p(x)-\sin(\pi/2)}{x-\pi/4}

. Interpret the result in the context of differentiability at

x=\frac{\pi}{4}

. Clearly state whether

p

is differentiable there and justify your conclusion.