Question

Find the asymptote of the graph of the function
$f\left(x\right)=\frac{2x^2+x-1}{x^2+x-2}$

Answer 1

Asymptote can be of two types: Horizontal and Vertical.

To find Horizontal asymptote of a rational function we need to check if the polynomial of numerator and denominator have same degree.

Here,the polynomial in numerator and denominator have same degree.

Thus, Horizontal aymptote:

$y=$ Co-efficient of highest degree term in numerator$÷$ Co-efficient of highest degree term in denominator

$\Rightarrow y=2/1$

$\Rightarrow y=2$

To find Vertical asymptote of a rational function we need to equate the denominator to zero.

$x^2+x-2=0\Rightarrow x^2+2x-x-2=0\Rightarrow x(x+2)-1(x+2)=0\Rightarrow (x+2)(x-1)=0$

Either $x+2=0\Rightarrow x=-2$ or $x-1=0\Rightarrow x=1$

Thus asymptotes of the given function are: $y=2,x=-2,x=1$