Question

How do you find the slant asymptote for \[\dfrac{{{x}^{3}}-1}{{{x}^{2}}-9}\]?

Answer 1

Hint: In the given question, we have been asked to find the asymptotes. In order to solve the question, first we need to start by defining the asymptote and the types of asymptotes. Later by applying the properties of asymptote we will apply different conditions and evaluating different values of ‘x’ and ‘y’. To find the slant asymptote, first we need to check that the degree of the numerator is greater than the denominator or not. If the highest degree of the numerator in the given expression is greater than the highest degree of the denominator in the given expression, thus slant asymptote occurs.

Complete step by step solution:
We have given that,
\[\dfrac{{{x}^{3}}-1}{{{x}^{2}}-9}\]
Let,
\[y=\dfrac{{{x}^{3}}-1}{{{x}^{2}}-9}\]
Factoring the denominator,
Using the property of numbers i.e. \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\]
Therefore,
\[y=\dfrac{{{x}^{3}}-1}{\left( x+3 \right)\left( x-3 \right)}\]
Finding the slant asymptote:
Since the highest degree of the numerator in the given expression is greater than the highest degree of the denominator in the given expression, thus slant asymptote occurs. i.e.
Degree of numerator > degree of denominator.
To finding the equation of slant asymptote;
We have to perform long polynomial division.
After performing the long polynomial division,
\[y=\dfrac{{{x}^{3}}-1}{{{x}^{2}}-9}\]
We have,
Quotient = \[x\]
Remainder = \[9x-1\]
Hence, the result is
\[y=\dfrac{{{x}^{3}}-1}{{{x}^{2}}-9}=x+\dfrac{9x-1}{{{x}^{2}}-9}\]
Thus,
The equation of slant asymptote is;
\[y=x\]
Therefore the slant asymptote of ‘y’ is \[x\].

Note: Students should always need to remember the concept of asymptote and very well know about the different types of conditions and their respective properties. Always make sure that all the terms are in the form of the same variable. If required in the given question always plot the graph of the asymptote and remember that first draw the axis then mark the point and plot the graph.