Rough introduction to infinite limits

Summary

• Suppose that $f$ is defined for all $x$ "near" $x_0$ (but not necessarily at $x_0$ ). We then say that $f(x)$ has the limit $∞$ as $x$ approaches $x_0$ and write

$\lim_{x→x_0}f(x)=∞$

• if $f(x)$ can be made arbitrarily large as $x$ tends to $x_0$ .
• $f(x)→-∞$ as $x→x_0$ is defined similarly.
• $f(x)→±∞$ as $x→±∞$ is defined similarly.
• Examples:
• $\frac{1}{x^2}→∞$ as $x→0$
• $logx→-∞$ as $x→0+$
• $\frac{1}{x}→∞$ as $x→0+$ while $\frac{1}{x}→-∞$ as $x→0-$
• $e^x→∞$ as $x→∞$
• Formal definition:
• Suppose that there is an open interval $(a,b)$ containing $x_0$ such that $f$ is defined for all $x$ in this interval possibly with the exception of $x_0$ . Then $f(x)$ approaches the limit $∞$ as $x$ approaches $x_0$ if for every $c$ , there exists a $δ>0$ such that $f(x)>c$ whenever $x_0-δ<x<x_0+δ$ .