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**2.2 Limits Involving Infinity**

Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

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**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

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**Adding 1 becomes insignificant as .**

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

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**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:

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Example 3: Find:

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**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.

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**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.

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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

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**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!

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**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:

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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

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

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

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