1. Ch. 2 – Limits and Continuity
2.2 – Limits Involving Infinity

2. Graph the following functions. Use the graph to find the desired limit.
Horizontal asymptotes give you the limits at ±∞!

3. Rational Function Review
Recall that rational functions (fractions with polynomials on top and bottom) often have horizontal asymptotes
Ex: Find the horizontal asymptotes of the following functions.
The degree of the top is smaller than the degree of the bottom
Horizontal asymptote: y = 0
The degree of the top equals the degree of the bottom
To find horizontal asymptote, divide leading coefficients: y = 3/2
The degree of the top is bigger than the degree of the bottom
Divide top by bottom to get slant asymptote

4. Horizontal asymptotes give you the limits at ±∞!
Ex: Find .
Use what you know about horizontal asymptotes…think of the graph!
Since f(x) has a vertical asymptote at y=0, f(x) will approach zero as x approaches infinity
Answer = 0
Since f(x) has a slant asymptote at y=0, f(x) will approach infinity as x approaches infinity
Answer = ∞
Horizontal asymptotes give you the limits at ±∞!

5. Some limits can be found as a result of several known limits!
Ex: Find algebraically.
(between -1 and 1)
(really big)

6. Ex: Find algebraically.
Ex: Find philosophically.
As x∞…
…the numerator stays between -1 and 1.
…the denominator becomes HUGE.
What’s 1 divided by a huge number?
Zero! So the answer is 0.

7. End Behavior Models
Some complicated functions can be modeled with a simpler function at extreme values of x (±∞)
Ex: Find power function end behavior models for the following functions:
As x approaches ±∞, the -4x3 dominates the other terms, so we model this function with f(x) = -4x3 .
As x approaches ±∞, the x2 terms dominate on top and bottom
Since the x2 terms cancel out, we model this function with f(x) = 3 .

8. Ex: Find left and right end behavior models for the function y = x2 + e-x .
Left-end behavior means as x-∞
Right-end behavior means as x∞
As x-∞, the e-x term dominates the function, so the left end model is f(x) = e-x .
As x∞, the x2 term dominates the function (e-∞ is zero), so the right end model is f(x) = x2 .