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 is said to be the vertical asymptote of a function , if either
Find the vertical asymptote of the graph of the function
Graph of the function
We are interested in finding the behavior of as . We observe that
Thus the line is the vertical asymptote of the given function.
Find the vertical asymptote of the graph of the function
Verifying the obtained asymptote with the help of a graph
We are interested in finding the behavior of as . We observe that
Similarly,
Thus the lines and are the vertical asymptotes for the given function.
Find the vertical asymptote of the graph of the function
Graph of the given function, verifying the asymptotes
We find that
Thus the line is the vertical asymptote of the given function.
Note: Observe that the graph also has an oblique asymptote on the line .