Question

How do you determine the vertical and horizontal asymptotes of the graph of the function?
$y=6-\dfrac{14}{4x+36}+\dfrac{1}{5{{x}^{4}}}$
What are the steps I need to do after I combine the three fractions together?

Answer 1

In this question we have been given with a function for which we have to find the vertical and horizontal asymptotes. We will first convert the expression into the form of a fraction with the same denominator. And because the expression will be a large fraction, we will find the vertical asymptotes by substituting the value of the denominator as zero. We will use the property of leading coefficients to find the horizontal asymptotes of the function and get the required solution.

Complete step-by-step solution:
We have the function given to us as:
$\Rightarrow y=6-\dfrac{14}{4x+36}+\dfrac{1}{5{{x}^{4}}}$
We can see that the function consists of three fractions therefore, we will simplify them to become a single fraction.
Since the term $6$ has no denominator, its denominator is $1$ therefore, the function can be written as:
$\Rightarrow y=\dfrac{6}{1}-\dfrac{14}{4x+36}+\dfrac{1}{5{{x}^{4}}}$
We will breakdown the method for finding the lowest common multiple in two part. On taking the lowest common multiple of the first two terms, we get:
$\Rightarrow y=\dfrac{6\left( 4x+36 \right)-14}{4x+36}+\dfrac{1}{5{{x}^{4}}}$
On simplifying, we get:
$\Rightarrow y=\dfrac{24x+216-14}{4x+36}+\dfrac{1}{5{{x}^{4}}}$
On simplifying, we get:
$\Rightarrow y=\dfrac{24x+202}{4x+36}+\dfrac{1}{5{{x}^{4}}}$
On taking the lowest common multiple of the remaining two terms, we get:
$\Rightarrow y=\dfrac{\left( 24x+202 \right)\left( 5{{x}^{4}} \right)+\left( 1 \right)\left( 4x+36 \right)}{\left( 4x+36 \right)\left( 5{{x}^{4}} \right)}$
On multiplying the terms, we get:
$\Rightarrow y=\dfrac{120{{x}^{5}}+1010{{x}^{4}}+4x+36}{20{{x}^{5}}+180{{x}^{4}}}$, which is the required function in the form of a fraction.
Now to find the vertical asymptote we will substitute the value of the denominator as zero.
On substituting, we get:
$\Rightarrow 20{{x}^{5}}+180{{x}^{4}}=0$
On taking the term $20{{x}^{4}}$ common, we get:
$\Rightarrow 20{{x}^{4}}\left( x+9 \right)=0$
Now we know the property that when $ab=0$ either $a=0$ or $b=0$ therefore, we get:
$\Rightarrow 20{{x}^{4}}=0$ and $x+9=0$
On rearranging, we get:
$\Rightarrow x=0$ and $x=-9$.
This implies that the vertical asymptote at $x=0$ and $x=-9$.
Now to find the horizontal asymptote, the biggest power of the numerator and denominator are to be considered.
$\Rightarrow y=\dfrac{120{{x}^{5}}+1010{{x}^{4}}+4x+36}{20{{x}^{5}}+180{{x}^{4}}}$
We can see that the highest exponent is $5$ and is the same for the numerator and the denominator.
Therefore, the fraction of their coefficients will give us the horizontal asymptote.
On making a fraction, we get:
$\Rightarrow y=\dfrac{120}{20}$
On simplifying, we get:
$\Rightarrow y=6$, which is the horizontal asymptote of the function.

Note: It is to be remembered that while calculating the horizontal asymptote the fraction of the coefficient of the largest exponent in the numerator and denominator is considered only when they are the same. If the numerator exponent is greater, then there exists no horizontal asymptote. If the numerator's exponent is smaller, then the horizontal asymptote is $y=0$.