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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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As the denominator gets larger (as x ), the overall fraction value gets smaller. There is a horizontal asymptote if either: or Limits outward toward infinity

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

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Example 2: Find: 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. If the denominator is positive, then the fractions overall sign is positive. If the denominator is negative, then the fractions overall sign is negative. vertical asymptote at x =0. The overall limit as x approaches zero does not exist, since the left-hand and right-hand limits dont match.

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

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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: dominant terms in numerator and denominator when |x| large f(x)

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You can adjust the direction of limits: outbound limit for an input equals inbound limit (toward x = 0) for the inputs reciprocal

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