Sometimes a function's values grow arbitrarily large (positive or negative) as $x$ approaches a particular number. This behavior is described using infinite limits, and it corresponds to a vertical asymptote on the graph.
let take function $\frac{1}{x^2}$ as an example, to find
$$ \lim_{x \rightarrow 0}\frac{1}{x^2} $$
As $x$ becomes close to $0$, $x^2$ also becomes close to $0$, and $1/x^2$ becomes very large.
In fact, it appears from the graph of the function $f(x)=\frac{1}{x^2}$ shows that the values of $f(x)$ can be made arbitrarily large by taking $x$ close enough to $0$. Thus the values of $f(x)$ do not approach a number, so $\lim_{x \rightarrow 0}\left(\frac{1}{x^2}\right)$ does not exist.
To indicate the kind of behavior exhibited in $\lim_{x \rightarrow 0}\frac{1}{x^2}$, we use the notation
$$ \lim_{x \rightarrow 0}\frac{1}{x^2}=\infty $$
This does not mean that we are regarding $\infty$ as a number. Nor does it mean that the limit exists. It simply expresses the particular way in which the limit does not exist: $\frac{1}{x^2}$ can be made as large as we like by taking $x$ close enough to $0$.
In general, we write symbolically
$$ \lim_{x \rightarrow a}f(x)=\infty $$
means that the values of $f(x)$ can be made arbitrarily large (as large as we please) by taking $x$ sufficiently close to $a$, but not equal to $a$.
To indicate that the values of $f(x)$ tend to become larger and larger (or “increase without bound”) as $x$ becomes closer and closer to $a$.
A similar sort of limit, for functions that become large negative as $x$ gets close to $a$, is illustrated.
$$ \lim_{x \rightarrow a}f(x)=-\infty $$
Means that the values of $f(x)$ can be made arbitrarily large negative by taking $x$ sufficiently close to $a$, but not equal to $a$.
Similar definitions can be given for the one-sided infinite limits
$$ \lim_{x \rightarrow a^-}f(x)=\infty $$
$$ \lim_{x \rightarrow a^-}f(x)=-\infty $$
$$ \lim_{x \rightarrow a^+}f(x)=\infty $$
$$ \lim_{x \rightarrow a^+}f(x)=-\infty $$
Remembering that $x \rightarrow a^-$ means that we consider only values of $x$ that are less than $a$, and similarly $x \rightarrow a^+$ means that we consider only $x>a$.
The vertical line $x = a$ is called a vertical asymptote of the curve $y=f(x)$ if at least one of the following statements is true: