Calculus Notes
⟵Section 1.4 Section 2.1 ⟶Section 1.5: Infinite Limits
If, as $x$ approaches a particular value $c$, the corresponding $f(x)$ becomes an arbitrarily large positive number, then $f(x)$ is said to have an infinite limit at $x=c$. This is written mathematically as $$\lim_{x\to c} f(x) = \infty$$ If $f(x)$ instead approaches an arbitrarily large negative number, then $$\lim_{x\to c} f(x) = -\infty$$Note that labeling the point as $\infty$ or $-\infty$ does not imply that the limit exists. Rather, an infinite limit fails to exist, because it does not converge to a specific value, as stated in the definition of a limit as given in Section 1.2.
A function that has an infinite limit at a particular point will also have a vertical asymptote at that point. In particular, if $g(c) = 0$ and $f(c)\ne 0$, then the function $h(x) = f(x)/g(x)$ has an infinite limit— and a vertical asymptote—at $x=c$.
⟵Section 1.4 Section 2.1 ⟶