2.2A Limits Involving Horizontal Infinity

1. 2.2A Limits Involving Horizontal Infinity Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

2. Limits outward “toward infinity” As the denominator gets larger (as x →∞), the overall fractionvalue gets smaller. There is a horizontal asymptote if either: or

3. Adding 1 becomes insignificant as . There is a horizontal asymptote at a height limit of 1. Example 1:

4. Find: When we graph this function, the limit appears to be zero. so for : by the Sandwich Theorem: Example 2:

5. Example 3: Find:

6. A function g(x) is: a right-end-behaviormodelfor f if and only if a left-end-behaviormodelfor f if and only if End Behavior Models An end behavior model describes the height behavior of a function as x approaches positive infinity or asx approachesnegative infinity.

7. As , approaches zero... becomes a right-end behavior model. As , is much further away from the x-axis, therefore is dominant on the left. becomes a left-end behavior model. Example 7: Find right- and left-end models for whereas x is further from the x-axis, so x dominates on the right. Test of model Test of model Our model is correct!

8. becomes a right-end behavior model. On your calculator, compare which pairs of graphs match the best on the right and on the left of the y-axis: Window: becomes a left-end behavior model. Example 7:

9. Example: For rational functions, the end behavior model comes from the highest power terms in the numerator and denominator: f(x) ≈ when |x| large dominant terms in numerator and denominator

10. You can adjust the direction of limits: “outbound” limit for an input equals “inbound” limit (toward x = 0) for the input’s reciprocal p