Question

How do you find vertical asymptotes using limits?

Answer 1

Hint:
Given an equation. We have to find the vertical asymptotes using the limits. First, we will apply the limits to the curve $f\left(x\right)$. Then, substitute the value of limit into the variable x and find the value of the function.

Complete step by step solution:
An asymptote is basically a line which the graph of a particular function approaches but never touches. A particular function has two types of asymptotes horizontal and vertical.
 The vertical line is defined as $x=k$ to the curve of the function $y=f\left(x\right)$.
Then, we will apply the limits to the function.
$\Rightarrow \underset{x\to k^{+}}{lim}f\left(x\right)$ and
$\Rightarrow \underset{x\to k^{-}}{lim}f\left(x\right)$
The vertical line $x=k$ is known as a vertical asymptote if either
$\Rightarrow \underset{x\to k^{+}}{lim}f\left(x\right)=\pm \mathrm{\infty }$ or
$\Rightarrow \underset{x\to k^{-}}{lim}f\left(x\right)=\pm \mathrm{\infty }$
Therefore, we have to determine the value of k such that either left hand limit or right hand limit will be $\pm \mathrm{\infty }$.
For example, consider a function $y={\displaystyle \frac{e^x}{x}}$
To find the vertical asymptotes of the function, apply the limits to the function,
$\Rightarrow \underset{x\to 0^{+}}{lim}{\displaystyle \frac{e^x}{x}}$ and
$\Rightarrow \underset{x\to 0^{-}}{lim}{\displaystyle \frac{e^x}{x}}$
Now, we will substitute the limit to the function.
$\Rightarrow \underset{x\to 0^{+}}{lim}{\displaystyle \frac{e^x}{x}}=\mathrm{\infty }$ and
$\Rightarrow \underset{x\to 0^{-}}{lim}{\displaystyle \frac{e^x}{x}}=\mathrm{\infty }$
Therefore, at $x=0$ the function $y={\displaystyle \frac{e^x}{x}}$ has a vertical asymptote.

Note:
The students please note that the vertical asymptotes are parallel to the y-axis when x approaches any constant value c, whereas the horizontal asymptotes are parallel to the x-axis. To find the vertical asymptotes of a rational function, we will set the denominator equal to zero and apply the limits to the expression. The students must remember that there are some functions for which the vertical asymptotes do not exist, such as the exponential function because exponent x may have any value.