# 2.5 | Infinite Limits

Let $$f$$ be a function defined on an open interval containing $$a$$ with the possible of $$a$$ itself. Then
$$\lim_{x \rightarrow a} f(x) = \infty$$
if all the values of $$f$$ can be made arbitrarily large (as large as we please) by taking $$x$$ sufficiently close to but not equal to $$a$$. Similarly,
$$\lim_{x \rightarrow a} f(x) = – \infty$$
if all the values of $$f$$ can be made as large in absolute value and negative as we please by taking $$x$$ sufficiently close to but not equal to $$a.$$
The line $$x = a$$ is a vertical asymptote of the graph of a function $$f$$ if at least one of the following statements is true:
$$\lim_{x\rightarrow a^-} f(x) = \pm \infty;$$
$$\lim_{x\rightarrow a^+} f(x) = \pm \infty;$$
$$\lim_{x\rightarrow a} f(x) = \pm \infty$$
Let $$f$$ be a function that is defined on an interval $$(a, – \infty).$$ Then the limit of $$f(x)$$ as $$x$$ approaches infinity (increases without bound) is the number $$L,$$ written
$$\lim_{x \rightarrow \infty} f(x) = L$$
if all the values of $$f$$ can be made arbitrarily close to $$L$$ by taking $$x$$ to be sufficiently large.
Let $$f$$ be a function that is defined on the interval $$(-\infty, a).$$ Then the limit of $$f(x)$$ as $$x$$ approaches negative infinity (decreases without bound) is the number $$L,$$ written
$$\lim_{x \rightarrow – \infty} f(x) = L$$
if all the values of $$f$$ can be made arbitrarily close to $$L$$ by taking $$x$$ to be sufficiently large in absolute value and negative.
The line $$y = L$$ is a horizontal asymptote of the graph of a function $$f$$ if
$$\lim_{x \rightarrow \infty} f(x) = L$$
or
$$\lim_{x \rightarrow – \infty} f(x) = L$$
(or both).
Let $$r > 0$$ be a rational number. Then
$$\lim_{x \rightarrow \infty} \frac{1}{x^r} = 0$$
Also, if $$x^r$$ is defined for all $$x,$$ then
$$\lim_{x \rightarrow – \infty} \frac{1}{x^r} = 0$$