What are the vertical and horizontal asymptotes of #g(x)=(x+7)/(x^2-4)#?

1 Answer
Feb 12, 2016

The horizontal asymptote is #y=0# and the vertical asymptotes are # x=2# and #x=-2#.


There are three basic rules for determining a horizontal asymptote. All of them are based off the highest power of the numerator (the top of the fraction) and the denominator (the bottom of the fraction).

If the numerator's highest exponent is larger than the denominator's highest exponents, no horizontal asymptotes exist. If the exponents of both top and bottom are the same, use the coefficients of the exponents as your y=.
For example, for #(3x^4)/(5x^4)#, the horizontal asymptote would be #y= 3/5#.

The last rule deals with equations where the denominator's highest exponent is larger than the numerator's. If this occurs, then the horizontal asymptote is #y=0#

To find the vertical asymptotes, you only use the denominator. Because a quantity over 0 is undefined, the denominator cannot be 0. If the denominator equals 0, there is a vertical asymptote at that point. Take the denominator, set it to 0, and solve for x.

#x^2-4= 0#
#x^2= 4#
#x=(+/-) 2#

x equals -2 and 2 because if you square both, they yield 4 even though they are different numbers.
Basic rule of thumb: If you square root a number, it is the positive and the negative quantity of the actual square root because the negative of the square root will produce the same answer as the positive when squared.