Chapter 2.6, Problem 81E

(a) Prove that

$\underset{x\to \infty }{\lim }f(x)=\underset{t\to 0^{+}}{\lim }f(1/t)$

and

$\underset{x\to -\infty }{\lim }f(x)=\underset{t\to 0^{-}}{\lim }f(1/t)$

if these limit exist.

(b) Use part (a) and Exercise 65 to find

$\underset{x\to 0^{+}}{\lim }x\sin \frac{1}{x}$