How do you find the horizontal asymptote for # g(x) = (3x^2) / (x^2 - 9)#?

Question

Benjamin Achtenberg · Accepted Answer

Solve for #x^2-9=0# In order to find the horizontal asymptote of a function, you have to solve it's denominator, while in order to find the vertical asymptote, you'll have to solve #y=a/c#, such that #y=(ax+b)/(cx+d)#. Note, the principles hold true for all degrees of #x#, so you can also use these two formulae if your function has a squared, or cubic, #x#. To find the horizontal asymptote of #g(x)#, solve the denominator part. Hence, #x^2-9=0# #x^2=9# #x^2=3^2# so, #x=3# Therefore the horizontal asymptote of #g(x)# is #x=3#, or #x-3=0#.

James Allen · Answer

undefined To find the horizontal asymptote of $ g(x) = \frac{3x^2}{x^2 - 9} $:
1. Compare the degrees of the numerator and denominator.
2. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $ y = 0 $.
3. If the degrees are equal, divide the leading coefficients to find the horizontal asymptote.
4. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
In this case, since both the numerator and denominator have the same degree (2), divide the leading coefficients: $ 3/1 = 3 $. So, the horizontal asymptote is $ y = 3 $.