Consider the function defined on the open interval .

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Consider the function

g(x)=\tan x

defined on the open interval

\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right)

. (a) Identify the complete set of real numbers at which

g

is differentiable, and provide a brief justification based on the analytic form of

g(x)

. (b) Explain why the derivative

g'(x)

becomes unbounded as

x

approaches

\tfrac{\pi}{2}

from the left. Your explanation should reference the behavior of elementary trigonometric functions in the neighborhood of

x=\tfrac{\pi}{2}