finding asymptotes and holes

Question

find the asymptotes and holes of x²+x-12/x²-9 and describe the behavior of the graph around the vertical asymptotes. 

Answer 1

When looking for discontinuities of a rational function, it is best to factor both the numerator and denominator as completely as possible. For this problem, you'll get

 

(x+4)(x-3) / (x+3)(x-3)

 

Zeros of the denominator indicate discontinuities of the function. In this case, x = 3 and x = -3 are discontinuities. To determine whether they are a hole or an asymptote, you have to compare them to the numerator.

 

-If the zero is in both the numerator and the denominator, it is a hole.

-If the zero is only in the denominator, it is an asymptote.

 

For this problem, x = 3 is a hole, and x = -3 is an asymptote.

 

To evaluate the behavior around the asymptote, plug in values slightly larger and smaller than the zero, and determine if they're positive or negative, then combine to determine the overall sign.

 

For x < -3,

 

(x+4)(x-3) / (x+3)(x-3)

(-3.1+4)(-3.1-3) / (-3.1+3)(-3.1-3)

(+)(-) / (-)(-)

and the result is negative. On the left side of the asymptote, the function goes toward -∞

 

For x > -3

 

(x+4)(x-3) / (x+3)(x-3)
(-2.9+4)(-2.9-3) / (-2.9+3)(-2.9-3)
(+)(-) / (+)(-)

and the result is positive. On the right side of the asymptote, the function goes toward +∞