Question

How do you find the vertical asymptote of a rational function?

Answer 1

In this question, we have to find out the vertical asymptote of a rational function to get the solution.
We know, $x = k$ is asymptote to the curve $y = f(x)$ if
$\mathop {\lim }\limits_{x \to {k^ + }} f(x) = + \infty$ or $- \infty \&$
$\mathop {\lim }\limits_{x \to {k^ - }} f(x) = + \infty$ or $- \infty$
So by putting the limit we will find the vertical asymptote. Finally we get the required answer.

Complete step-by-step solution:
We need to evaluate the steps for finding vertical asymptote of a rational function.
An asymptote is a line that the graph of a function approaches but never touches.
Vertical asymptote:
We know $x = k$ is asymptote to the curve $y = f(x)$ if
$\mathop {\lim }\limits_{x \to {k^ + }} f(x) = + \infty$ or $- \infty \&$
$\mathop {\lim }\limits_{x \to {k^ - }} f(x) = + \infty$ or $- \infty$
Now we have to discuss the vertical asymptotes that are found when the function is not defined. Here, the denominator must be 0 for this to occur.
Also, we can defined Rational functions are those which can be written in $\dfrac{{p\left( x \right)}}{{q\left( x \right)}}$ format, where both the numerator and denominator are functions of$x$.
To find the vertical asymptotes of a rational function, we need to simply equate the denominator equals to zero and solve for $x$.
Hence, we have to find the vertical asymptote of a rational function; we just need to equate the denominator function equals to zero and by solving that we will get the vertical asymptote.

Note: A straight line is said to be an asymptote to the curve $y = f\left( x \right)$ if the distance of point $P\left( {x,{\text{ }}y} \right)$ on the curve from the line tends to zero when $x \to \infty$ or $y \to \infty$ both $x$ &$y \to \infty$.
Rational function:
Rational functions are functions, which are created by dividing two functions.
Formally, they are represented as $\dfrac{{f\left( x \right)}}{{g\left( x \right)}}$, where $f\left( x \right),g\left( x \right)$ are both functions.