Rough introduction to limits at infinity

Summary

• Suppose that $f$ is defined for arbitrary large $x$ -values. We then say that $f(x)$ approaches the limit $L$ as $x$ tends to infinity and write

$\lim_{x→∞}f(x)=L$

• if $f(x)$ can be made arbitrarily close to $L$ by making $x$ sufficiently large.

$\lim_{x→-∞}f(x)=L$

• is defined similarly
• Examples:
• $\frac{1}{x}→0$ as $x→∞$
• $e^x→0$ as $x→-∞$
• Formal definition:
• Suppose that $f$ is defined for all $x$ in $(a,∞)$ for some $a$ . Then $f(x)$ approaches the limit $L$ as $x$ approaches $∞$ if for every number $ε>0$ , there exists a $c$ such that $L-ε<f(x)<L+ε.$ whenever $x>c$ .