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2.5 Limits Involving Infinity

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1 2.5 Limits Involving Infinity
Greg Kelly, Hanford High School, Richland, Washington

2 As the denominator gets larger, the value of the fraction gets smaller.
There is a horizontal asymptote if: or

3 This number becomes insignificant as .
Example 1: This number becomes insignificant as There is a horizontal asymptote at 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 Infinite Limits: As the denominator approaches zero, the value of the fraction gets very large. vertical asymptote at x=0. If the denominator is positive then the fraction is positive. If the denominator is negative then the fraction is negative.

7 The denominator is positive in both cases, so the limit is the same.
Example 4: The denominator is positive in both cases, so the limit is the same.

8 End Behavior Models: End behavior models model the behavior of a function as x approaches infinity or negative infinity. A function g is: a right end behavior model for f if and only if a left end behavior model for f if and only if

9 becomes a right-end behavior model.
Example 7: As , approaches zero. (The x term dominates.) becomes a right-end behavior model. Test of model Our model is correct. As , increases faster than x decreases, therefore is dominant. 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, graph: Use:

11 Right-end behavior models give us:
Example 7: Right-end behavior models give us: dominant terms in numerator and denominator

12 Often you can just “think through” limits.
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