1. 3.5: ASYMPTOTES

2. VERTICAL ASYMPTOTES
If f (x) approaches infinity (or negative infinity) as x approaches c from the right or from the left, then the line x = c is a vertical asymptote of the graph of f.

3. Look at the graph of

4. EXAMPLES
Evaluate the following limits:

5. MORE THAN ONE ASYMPTOTE
Find the vertical asymptote(s) of the graph of

6. EXAMPLE Find the vertical asymptote(s) of the graph of
What happens at x = -3 ?

7. EXAMPLE
Find the limits:

8. HORIZONTAL ASYMPTOTES
If f is a function and L1 and L2 are real numbers, the statements below denote limits at infinity. The lines y = L1 and y = L2 are horizontal asymptotes of the graph of f.

9. FIND THE LIMIT:

10. HORIZONTAL ASYMPTOTES
Let f (x) = p (x)/q (x) be a rational function.
If the degree of the numerator is less than the degree of the denominator, then y = 0 is a horizontal asymptote of the graph of f.
If the degree of the numerator is equal to the degree of the denominator, then y = a/b is a horizontal asymptote of the graph of f, where a and b are the leading coefficients of p(x) and q(x), respectively.
If the degree of the numerator is greater than the degree of the denominator, then the graph of f has no horizontal asymptote.

11. FIND THE HORIZONTAL ASYMPTOTE OF THE GRAPH OF THE FUNCTION: