Define the function q(x)= cases (3x)-3x x^3 , & x 0, , & x=0

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Define the function

q(x)=\begin{cases}\dfrac{\tan(3x)-3x}{x^3}, & x\ne 0,\\d, & x=0,\end{cases}

where

d

is a real constant. (a) Show that

\displaystyle\lim_{x\to 0}q(x)

exists and compute its value. (b) Determine the value of

d

that removes any discontinuity at

x=0

. (c) Prove that the extended function is not only continuous but also differentiable at

x=0

. Carefully compute

(q)^{\,'}(0)