Question

How do you find the vertical, horizontal or slant asymptotes for \[\dfrac{1}{{{x^2} + 4}}?\]

Answer 1

Hint: As we know that asymptote of a curve is the lien formed by the movement of the curve and the line moving continuously towards zero. We know that if the degree of the denominator is greater than the degree of the numerator, horizontal asymptote is at $y = 0$. In this question, to find the value of the vertical, horizontal or slant asymptotes we see the degree of the denominator and equate to zero.

Complete step by step solution:
First we will find the value of the vertical asymptotes: This occurs when the rational function contains a zero denominator. So we can write it as ${x^2} + 4 = 0$.
We will solve it now, ${x^2} = - 4 \Rightarrow x = \sqrt { - 4} $.
As we can see that there are no real solutions so there are no vertical asymptotes of this function.
In the horizontal asymptotes the value occurs if the numerator of a rational function has degree less than or equal to the degree of the denominator. But here this is not the case,
We can see that the numerator degree is only one and denominator degree is two, so the horizontal asymptote is $y = 0$.
In the slant asymptotes, the same rule applies, as the numerator is lower degree than the denominator. So there are no slant asymptotes.
Hence the answer of vertical asymptotes is that there are no value as it has no real solutions, while horizontal asymptotes is $y = 0$.

Note: We should note that for finding the slant asymptotes we have to use the concept that it occurs when the degree of the denominator of a rational function is one less than the degree of the numerator. There are three forms of asymptotes: vertical, horizontal and oblique. It is noted that if the degree of the denominator is $ > $ degree of the numerator, then the horizontal asymptote is $y = 0$.