Infinite limit

Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a$ be a limit point of $D$. The notation $f(x) \to +\infty$ as $x \to a$ means that for each real $M$ there is $\delta > 0$ with $f(x) > M$ for all $x \in D$ with $0 < |x - a| < \delta$. The definition of $f(x) \to -\infty$ is the same, with $f(x) < M$.

Remarks

An infinite limit is not a limit in the sense of limit of a function, which requires a real value $L$. For example, $1/x^2 \to +\infty$ as $x \to 0$. The same idea gives $f(x) \to +\infty$ as $x \to +\infty$. An example is $f(x) = x^2$.

Explanations

L1 human

An infinite limit means that $f(x)$ becomes as large as we like when $x$ is close to $a$. The graph then has a vertical asymptote at $x = a$.