Mathematics

Problem

Consider the function

g

given by

g(x)=\lvert x\rvert\,\sin\!\left(\tfrac{\pi}{x}\right)\quad (x\ne0).

Note that

g

is not defined at

x=0

. Answer the following questions carefully. (a) Determine

\displaystyle \lim_{x\to0^-}g(x)

or state why it does not exist. Support your claim with appropriate reasoning based on the behavior of

\sin\!\left(\tfrac{\pi}{x}\right)

. (b) Determine

\displaystyle \lim_{x\to0^+}g(x)

or state why it does not exist. Support your claim. (c) Using your answers to parts (a) and (b), decide whether

\displaystyle \lim_{x\to0}g(x)

exists. Justify fully.