How do you find the asymptotes for #f(x)=(3x^2)/(x^2+3)#?

The rules for horizontal asymptotes:

• When the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
• When the degree of the numerator is the same as the degree of the denominator, there is a horizontal asymptote at #x=0#
• When the degree of the numerator is less that the degree of the
  denominator, there is a horizontal asymptote at the quotient of the leading coefficients.

Because the degrees in this equation are the same, there is a horizontal asymptote at #x=0#

The rule for vertical asymptotes:

• There is a vertical asymptote at any value that will cause the function to be undefined.

Because no values will cause the function to be undefined, there are no vertical asymptotes.

The rule for vertical asymptotes:

• When the degree of the numerator is exactly #1# more than the degree of the denominator, there is a slant asymptote at the quotient of the numerator and the denominator. (You have to divide the top by the bottom)

Because the degrees are the same, there is no slant asymptote.