Vertical Asymptotes

An asymptote is a horizontal/vertical oblique line whose distance from the graph of a function keeps decreasing and approaches zero but never gets there.

Asymptotes have a variety of applications: they are used in big O notation, they are simple approximations to complex equations, and they are useful for graphing rational equations. In this wiki, we will see how to determine the vertical asymptote of a given curve.

A line $x=c$ is said to be the vertical asymptote of a function $y=f(x)$, if either

$\lim_{x\rightarrow c^+}f(x)=\pm \infty \quad\text{or}\quad\displaystyle\lim_{x\rightarrow c^-}f(x)=\pm \infty.$

Find the vertical asymptote of the graph of the function

$f(x)=\dfrac{3}{x-2}.$

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Graph of the function

We are interested in finding the behavior of $f(x)$ as $x\rightarrow 2$. We observe that

$\displaystyle\lim_{x\rightarrow 2^+}\dfrac{3}{x-2}=\infty\quad\text{and}\quad \displaystyle\lim_{x\rightarrow 2^-}\dfrac{3}{x-2}=- \infty.$

Thus the line $x=2$ is the vertical asymptote of the given function. $_\square$

Find the vertical asymptote of the graph of the function

$f(x)=\dfrac{4}{x^2-25}.$

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Verifying the obtained asymptote with the help of a graph

We are interested in finding the behavior of $f(x)$ as $x\rightarrow \pm 5$. We observe that

$\displaystyle\lim_{x\rightarrow 5^+}\dfrac{4}{x^2-25}=\infty\quad\text{and}\quad \displaystyle\lim_{x\rightarrow 5^-}\dfrac{4}{x^2-25}=- \infty.$

Similarly,

$\displaystyle\lim_{x\rightarrow -5^+}\dfrac{4}{x^2-25}=-\infty\quad\text{and}\quad \displaystyle\lim_{x\rightarrow -5^-}\dfrac{4}{x^2-25}=\infty.$

Thus the lines $x=5$ and $x=-5$ are the vertical asymptotes for the given function. $_\square$

Find the vertical asymptote of the graph of the function

$f(x)=\dfrac{x^2+4}{x-3}.$

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Graph of the given function, verifying the asymptotes

We find that

$\displaystyle\lim_{x\rightarrow 3^+}\dfrac{x^2+4}{x-3}=\infty\quad\text{and}\quad\displaystyle\lim_{x\rightarrow 3^-}\dfrac{x^2+4}{x-3}=-\infty.$

Thus the line $x=3$ is the vertical asymptote of the given function. $_\square$

Note: Observe that the graph also has an oblique asymptote on the line $y=x+3$.