1. Limits at Infinity
Section 3.5 AP Calc

2. Find the end behavior by completing the table:
Given
Find the end behavior by completing the table: x
-100
-10 -1 1 10
100
f(x)

3. Definition of Limits at Infinity
Let L be a real number:
1) The statement means for
each there exists an M > 0 such that
wherever x > M.

4. 2) The statement means for
each there exists an N>0 such that
wherever x<N.

5. Definition of Horizontal Asymptote
The line y=L is a horizontal asymptote of the graph of f if or

6. Properties (If f and g both have a limit that exists):

7. Find the limit:

8. Thm 3.10 Limits at Infinity
If r is a positive rational number and c is a real number, then
Furthermore, if xr is defined when x<0, then

9. Find the limit:

10. Find the limits:

11. Guidelines for finding Limits at Infinity of Rational Functions:
1. If degree numerator less than degree denominator, limit 0.
2. If degree num. equal to degree den., limit is ratio of leading coefficients.
3. If degree num. greater then degree den., limit does not exist.

12. Find the limit:

13. Find the limit:

14. Definition Infinite Limits at Infinity
Let f be a function defined on the interval (a,∞).
1. The statement means for each positive number M, there is a corresponding number N>0 such that f(x)>M wherever x>N.
2. The statement means for each negative number M, there is a corresponding number N>0 such that f(x)<M wherever x>N.

15. Find the limit: