Section 2.6 Limits at Infinity, Horizontal Asymptotes "Being defeated is often a temporary condition. Giving up is what makes it permanent." Marilyn vos Savant (1946-)

As x approaches +/-infinity ,the graph of f(x) can either Approach zero Grow without bounds (either up or down) Approach some other number Continually fluctuate

The Great Battle!

The Great Battle! The “degree” of the functions in a rational expression will determine the end behavior of the function. If Top > Bottom, then If Bottom > Top If Bottom = Top Watch out for radicals!

Definition Let f be a function defined on some interval Means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. Definition Means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large negative. Let f be a function defined on some interval

Definition The line is called a horizontal asymptote of the curve if either Or

Example 1 Find the infinite limits, limits at infinity and asymptotes for the function f shown here.

Example 2 Find

1 Evaluate the limit. A 0 B Infinity C Negative Infinity D 1 E DNE

2 Evaluate the limit. A 0 B Infinity C Negative Infinity D 1 E DNE

3 Evaluate the limit. A 0 B Infinity C Negative Infinity D 3/5 E DNE

4 Evaluate the limit. A 0 B Infinity C Negative Infinity D 1 Warning: Algebra ahead! E DNE

5 Evaluate the limit. A 0 B Infinity C Negative Infinity D +/-1 E DNE

6 Evaluate the limit. A 0 B Infinity C Negative Infinity D +/-1 E DNE

7 Evaluate the limit. A 0 B Infinity C Negative Infinity D +/-1 E DNE

8 Evaluate the limit. A 0 B Infinity C Negative Infinity D +/-1 E DNE

9 Evaluate the limit. A 0 B Infinity C Negative Infinity D +/-1 E DNE

Theorem If is a rational number, then is a rational number such that is defined for all x, then If

Example 4 Find the horizontal and vertical asymptotes of the graph of the function

Example 11 Sketch the graph of by finding its intercepts and its limits as and