Mathematics

Problem

Functions

f

and

g

are defined for all real

x

f(x)=\begin{cases} x^2\sin\!\bigl(\tfrac1x\bigr), & x\neq 0,\\[4pt] 0, & x=0, \end{cases} \qquad g(x)=\begin{cases} x\,e^{x^2}, & x\neq 0,\\[4pt] 0, & x=0. \end{cases}

Investigate the limit

\displaystyle \lim_{x\to 0}\;\dfrac{f(x)}{g(x)}

by answering the following. a. Show that both

f(x)

and

g(x)

approach

0

x\to 0

, and hence the quotient is a candidate for an indeterminate ratio of type

0/0

. Support your answer with appropriate limit computations or squeeze-theorem arguments. b. Determine whether

f

is differentiable on an open interval containing

x=0

. Justify your conclusion with rigorous reasoning (a formal derivative computation is not required away from

x=0

). c. Determine whether

g

is differentiable on an open interval containing

x=0

. Justify your conclusion. d. Based on your results in parts (b) and (c), explain whether substituting the derivatives of numerator and denominator for

f

and

g

near

x=0

would be a justifiable analytic maneuver for this limit problem.

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