Question

How do you find the vertical asymptotes of the function:
$y = \dfrac{{{x^2} + 1}}{{3x - 2{x^2}}}$ ?

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.

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

Hence the vertical asymptotes are $x = 0\,and\,x = \dfrac{3}{2}$.

Additional Information:
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.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 occur 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 occur when the degree of the denominator of a rational function is one less than the degree of the numerator.

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.