How do you find the vertical, horizontal or slant asymptotes for #y=(2x^2 + 3)/(x^2 - 6 )#?

Question

Serenity Johnson · Accepted Answer

vertical asymptotes # x = ± sqrt6#
horizontal asymptote y = 2

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation/s equate the denominator to zero. solve : # x^2 - 6 = 0 → x^2 = 6 → x = ± sqrt6# Horizontal asymptotes occur as #lim_(x→±∞) f(x) → 0# divide all terms on numerator / denominator by # x^2 # #(2x^2+3)/(x^2-6) =( (2x^2)/x^2 + 3/x^2)/(x^2/x^2 - 6/x^2)=(2+3/x^2)/(1-6/x^2)# now as x →∞ # , 3/x^2 " and " 6/x^2 → 0 # # rArr y = 2 " is the asymptote "# Slant asymptotes occur when the degree of the numerator is greater than the degree of the denominator. This is not the case here hence no slant asymptotes. Here is the graph of the function. graph{(2x^2+3)/(x^2-6) [-10, 10, -5, 5]}

Elias Ackerman · Answer

undefined To find the vertical asymptotes of the function $ y = \frac{2x^2 + 3}{x^2 - 6} $, you need to identify the values of $ x $ for which the denominator equals zero, as vertical asymptotes occur where the function is undefined.

To find horizontal or slant asymptotes, you compare the degrees of the numerator and the denominator polynomials:

1. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $.
2. If the degree of the numerator equals the degree of the denominator, divide the leading coefficients to find the horizontal asymptote.
3. If the degree of the numerator is greater than the degree of the denominator by exactly one, there's a slant asymptote. You can find it by performing polynomial long division or using synthetic division.

In your function $ y = \frac{2x^2 + 3}{x^2 - 6} $:

1. The denominator equals zero when $ x^2 - 6 = 0 $. Solve for $ x $ to find the vertical asymptotes.
2. Compare the degrees of the numerator and denominator to determine the presence of horizontal or slant asymptotes.

Perform these steps to identify the asymptotes for the given function.