How to find the asymptotes for #(3x^2) / (x^2-4)#?

Question

Sadie Alexander · Accepted Answer

#(3x^2)/(x^2-4)# has horizontal asymptote #y=3# and vertical asymptotes #x=-2# and #x=2#

Note that:

#lim_(x->+-oo) (3x^2)/(x^2-4) = lim_(x->+-oo) 3/(1-4/x^2) = 3#

So #y=3# is a horizontal asymptote.

Also:

#x^2-4 = (x-2)(x+2)#

So the denominator is zero when #x=+-2# and the numerator is non-zero at those values.

So #(3x^2)/(x^2-4)# has vertical asymptotes at #x=-2# and #x=2#

graph{(y-(3x^2)/(x^2-4)) = 0 [-10.09, 9.91, -3.6, 6.4]}

Christian Antunez · Answer

undefined To find the asymptotes of the function $ \frac{3x^2}{x^2-4} $, you need to consider the behavior of the function as $ x $ approaches certain values.

1. Vertical asymptotes: These occur where the denominator of the rational function equals zero. Set the denominator equal to zero and solve for $ x $. The values of $ x $ obtained are the vertical asymptotes.

2. Horizontal asymptotes: These occur when the degree of the numerator is less than or equal to the degree of the denominator. If the degrees are equal, divide the leading coefficients of both the numerator and denominator to find the horizontal asymptote.

In the case where the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote, but there may be a slant asymptote.

So, to find the asymptotes for $ \frac{3x^2}{x^2-4} $, follow these steps to find both vertical and horizontal asymptotes.