Define the functions u(x)= cases (2x)-2x,&x<0

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

Problem

Define the functions

u(x)=\begin{cases}\sin(2x)-2x,&x<0,\\ x^3\ln(1+x),&x\ge 0,\end{cases}\qquad v(x)=\begin{cases}1-\cos x-x^2/2,&x<0,\\ x\tan^{-1}x, &x\ge 0.\end{cases}

Let

H(x)=\dfrac{u(x)}{v(x)}

for

x\ne 0

. (a) Show that both

u(x)

and

v(x)

approach

0

x\to 0^-

. Then determine

\displaystyle\lim_{x\to 0^-}H(x)

. Provide full justification. (b) Show that both

u(x)

and

v(x)

approach

0

x\to 0^+

. Then determine

\displaystyle\lim_{x\to 0^+}H(x)

. Provide full justification. (c) Decide whether

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

exists. Explain your reasoning thoroughly. (d) Suppose instead that

H

is extended by defining

H(0)=L

, where

L

is any real number. Determine all possible values of

L

for which

H

is continuous at

x=0

. Justify your answer.