For Exercise, determine an equation for the horizontal asymptote for the graph of f. Consu...

For Exercise, determine an equation for the horizontal asymptote for the graph of f. Consult your work performed in Exercises 1–8.

In Exercises 1–8, determine the indicated limit algebraically using the method in Example 1. Verify your result numerically.

Exercises 1–8

1.

2.

3.

4.

5.

6.

7.

8.

Example 1

Determining Horizontal Asymptotes

For , determine , and , and determine the horizontal asymptote for the graph of f.

Perform the Mathematics

Because we have a rational function, the algebraic manipulation for problems of this type is rather straightforward:

Divide every term in the numerator and denominator by xb, where b is the larger degree of the numerator and the denominator.

For this problem, since both degrees are 2, we have b = 2. We divide every term in the numerator and denominator by x2. Th is gives

So, to determine the limit algebraically, we have

The algebraic determination for is identical. We immediately conclude that the horizontal asymptote is the line .