Question

How do you identify the oblique asymptote of $f(x) = \dfrac{{9 - {x^3}}}{{3{x^2}}}?$

Answer 1

Hint : To find the oblique asymptote of the given function, divide the numerator with the denominator and then put the limit of the variable tends to infinity to get the required oblique asymptote.

Complete step by step solution:
In order to find the oblique asymptote of the given function
$f(x) = \dfrac{{9 - {x^3}}}{{3{x^2}}}$ , we will first divide the numerator with the denominator as follows
$f(x) = \dfrac{{9 - {x^3}}}{{3{x^2}}} \\\ = \dfrac{9}{{3{x^2}}} - \dfrac{{{x^3}}}{{3{x^2}}} \\\ = \dfrac{3}{{{x^2}}} - \dfrac{x}{3} \;$
Now, if we go towards positive or negative infinity then the first term will tend to zero, so we will
neglect that term, because it will have a very smaller value
Therefore $y = - \dfrac{x}{3}$ is the required oblique asymptote of the given curve equation.
So, the correct answer is “ $y = - \dfrac{x}{3}$ ”.

Note : Oblique asymptotes are also known as slant asymptotes, because of their slanted form which represents a linear equation graph, $y = mx + c$ . A rational function will contain an oblique function only when its degree of numerator is greater than the degree of the denominator, as we can see in the question the degree of the numerator is three whereas the degree of the denominator is two which is less than three. Also when you see the graph of the given function with its oblique function then you will get to know that oblique function tells the end behavior of a function.