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
\[\lim_{x\rightarrow c^+}f(x)=\pm \infty \quad\text{or}\quad\displaystyle\lim_{x\rightarrow c^-}f(x)=\pm \infty. \ _\square\]
Find the vertical asymptote of the graph of the function
\[f(x)=\dfrac{3}{x-2}.\]
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. \(_\square\)
Graph of the function
Find the vertical asymptote of the graph of the function
\[f(x)=\dfrac{4}{x^2-25}.\]
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. \(_\square\)
Verifying the obtained Asymptote with the help of a graph
Find the vertical asymptote of the graph of the function
\[f(x)=\dfrac{x^2+4}{x-3}.\]
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. \(_\square\)
Graph of the given function, verifying the asymptotes
Note: Observe that the graph also has an oblique asymptote on the line \(y=x+3\).
