A function and its graph are given. Use the graph to find the horizontal asymptotes, if they exist. Confirm your results analytically.
E) no horizontal asymptotes

Question

A function and its graph are given. Use the graph to find the horizontal asymptotes, if they exist. Confirm your results analytically. ​  ​  ​
E) no horizontal asymptotes

Quizplus · Accepted Answer

From the graph, as x approaches infinity, the graph approaches y=3. As x approaches negative infinity, the graph approaches y=-1. 
Analytically, we can find the horizontal asymptotes by looking at the degrees of the numerator and denominator. In this case, the degree of the numerator (1) is less than the degree of the denominator (2). Therefore, there is a horizontal asymptote at y=0. However, this does not match what we see from the graph. We need to investigate further. 
We can use polynomial long division to simplify the function:
(x^2-4)/(x^2+1) = 1 - 4/(x^2+1)
As x approaches infinity, 4/(x^2+1) approaches 0, so the function approaches 1. As x approaches negative infinity, 4/(x^2+1) also approaches 0, so the function approaches 1. Therefore, there is a horizontal asymptote at y=1, which confirms our answer of choice C.

A Function and Its Graph Are Given