Problem

Find the horizontal and vertical asymptotes of

Solution

Possible vertical asymptotes are the places where f(x) is not defined; that is, where the denominator of f(x) is 0. This happens at x = 1 and x = –2. To determine if these lines are vertical asymptotes, we examine the limit of f(x) as x approaches these values.

As x approaches 1 from above, the numerator approaches 4, the term x + 2 in the denominator approaches 3, and the term x – 1 becomes a very small positive number. Therefore

As x approaches 1 from below, the numerator approches 4, the term x + 2 in the denominator approaches 3, and the term x – 1 becomes a negative number, very close to zero. Therefore

As x approaches –2 from above, the numerator approaches 16, the term x – 1 in the denominator approaches –3, and the term x + 2 becomes a very small positive number. Therefore

Finally, as x approaches –2 from below, the numerator approches 16, the term x – 1 in the denominator approaches –3, and the term x +2 becomes a negative number, very close to zero. Therefore

So the lines x = 1 and x = –2 are indeed vertical asymptotes.

To find horizontal asymptotes, we look at what happens when x approaches ¥ and –¥. Rewrite the function f(x) by expanding the denominator:

Because the polynomials in the numerator and the denominator are both of degreee 2, we use the Dominant Term Rule, which tells us that the limits

and

can be found by looking at the quotients of the highest powers of x in f(x). So:

So the line y = 4 is a horizontal asymptote.

Here's the graph of f(x) and its asymptotes.