Question

How do you find the vertical, horizontal and slant asymptotes of:
$\dfrac{{7x - 2}}{{{x^2} - 3x - 4}}$ ?

Answer 1

For finding the vertical asymptotes we have to use the concept that Vertical asymptotes occur at the values where a rational function contains a zero denominator.For finding the horizontal asymptotes we have to use the concept that it occurs when the numerator of a rational function has degree less than or equal to the degree of the denominator.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.

Complete step by step answer:
We have, $\dfrac{{7x - 2}}{{{x^2} - 3x - 4}}$.We have to find vertical, horizontal and slant asymptotes,
Vertical asymptotes: Vertical asymptotes occur at the values where a rational function contains a zero denominator.Therefore, solve ${x^2} - 3x - 4 = 0$ ,
${x^2} - 3x - 4 = 0 \\\ \Rightarrow (x - 4)(x + 1) = 0 \\\ \Rightarrow x = - 1,x = 4$
We have the required vertical asymptotes.

Horizontal Asymptotes: Horizontal Asymptotes occur when the numerator of a rational function has degree less than or equal to the degree of the denominator.
$\mathop {\lim }\limits_{x \to \pm \infty } f(x) \to c$
divide terms on numerator and denominator by ${x^2}$,
we will get ,
$\dfrac{{7x - 2}}{{{x^2} - 3x - 4}} \\\ \Rightarrow \dfrac{{\dfrac{{7x}}{{{x^2}}} - \dfrac{2}{{{x^2}}}}}{{\dfrac{{{x^2}}}{{{x^2}}} - \dfrac{{3x}}{{{x^2}}} - \dfrac{4}{{{x^2}}}}} = \dfrac{{\dfrac{7}{x} - \dfrac{2}{{{x^2}}}}}{{1 - \dfrac{3}{x} - \dfrac{4}{{{x^2}}}}} \\\$
As $x \to \pm \infty$,
$f(x) = \dfrac{{0 - 0}}{{1 - 0 - 0}} = 0 \\\ \Rightarrow y = 0$
We have the required horizontal asymptote.

Slant asymptotes:
Slant asymptotes occur when the degree of the denominator of a rational function is one less than the degree of the numerator.This is not the case here (numerator-degree is one and denominator-degree is two ). Hence there are no slant asymptotes.

Note: Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily. An asymptote could be a line that the graph of a function approaches as either ‘x’ or ‘y’ attend positive or negative infinity. There are three forms of asymptotes: vertical, horizontal and oblique.