Download presentation

1
**2.2 Limits Involving 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 fraction value gets smaller. There is a horizontal asymptote if either: or

3
**Adding 1 becomes insignificant as .**

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

4
**When we graph this function, the limit appears to be zero.**

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

5
Example 3: Find:

6
**Limits that are infinite in height**

As the denominator approaches zero, the magnitude of the fraction gets very large. vertical asymptote at x=0. If the denominator is positive, then the fraction’s overall sign is positive. If the denominator is negative, then the fraction’s overall sign is negative. The overall limit as x approaches zero does not exist, since the left-hand and right-hand limits don’t match.

7
**The overall limit as x approaches zero is **

Example 4: The denominator is positive after squaring x, so the limit has the same sign from left and right. The overall limit as x approaches zero is “positive without bound,” since the left-hand and right-hand limits both trend toward positive infinity.

8
End Behavior Models An end behavior model describes the height behavior of a function as x approaches positive infinity or as x approaches negative infinity. A function g(x) is: a right-end-behavior model for f if and only if a left-end-behavior model for f if and only if

9
**Find right- and left-end models for**

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

10
**becomes a right-end behavior model.**

Example 7: becomes a right-end behavior model. becomes a left-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:

11
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

12
**You can adjust the direction of limits:**

“outbound” limit for an input equals “inbound” limit (toward x = 0) for the input’s reciprocal p

Similar presentations

OK

Rational Functions A rational function is a function of the form where g (x) 0.

Rational Functions A rational function is a function of the form where g (x) 0.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on samuel taylor coleridge Ppt on schottky diode testing Ppt on polynomials for class 10 Download ppt on transportation in plants Ppt on human induced disasters Ppt on image compression using wavelets Ppt on nutrition in plants of class 7th Ppt on hindu religion holidays Ppt on cse related topics about psychology Ppt on mauryan dynasty