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

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

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

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:

Example 3: Find:

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.

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.

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

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!

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:

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

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