Lebl-Basic-Analysis-I · Problem 3.5.2

Continuous Functions

Mathematics · limits-at-infinity

Problem

Let

f : [1, \infty) \to \mathbb{R}

be a function. Define

g : (0, 1] \to \mathbb{R}

via

g(x) := f(1/x)

. Using the definitions of limits directly, show that

\lim_{x \to 0^+} g(x)

exists if and only if

\lim_{x \to \infty} f(x)

exists, in which case they are equal.