Functions and are defined for all real by f(x)= cases x^2 ! ( 1x ), & x 0, [4pt] 0, & x=0

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

Problem

Functions ff and gg are defined for all real xx by f(x)={x2sin⁡ ⁣(1x),x≠0,0,x=0,g(x)={x ex2,x≠0,0,x=0.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 lim⁡x→0  f(x)g(x)\displaystyle \lim_{x\to 0}\;\dfrac{f(x)}{g(x)} by answering the following. (1) Show that both f(x)f(x) and g(x)g(x) approach 00 as x→0x\to 0, and hence the quotient is a candidate for an indeterminate ratio of type 0/00/0. Support your answer with appropriate limit computations or squeeze-theorem arguments. (2) Determine whether ff is differentiable on an open interval containing x=0x=0. Justify your conclusion with rigorous reasoning (a formal derivative computation is not required away from x=0x=0). (3) Determine whether gg is differentiable on an open interval containing x=0x=0. Justify your conclusion. (4) Based on your results in parts (b) and (c), explain whether substituting the derivatives of numerator and denominator for ff and gg near x=0x=0 would be a justifiable analytic maneuver for this limit problem.

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