Question

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

Answer 1

Hint: 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.