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Explore the concept of limits involving infinity through engaging examples and clear explanations. This tutorial covers horizontal and vertical asymptotes, illustrating how fractions behave as the denominator grows larger or approaches zero. Learn how to identify end behavior models for functions, understand the significance of dominant terms in both the numerator and denominator, and analyze limits through the sandwich theorem. Join us as we graph functions to visualize how limits behave near critical points, ensuring a thorough grasp of this essential calculus topic.
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Photo by Vickie Kelly, 2006 Greg Kelly, Hanford High School, Richland, Washington 2.2 Limits Involving Infinity North Dakota Sunset
As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if: or
This number becomes insignificant as . There is a horizontal asymptote at 1. Example 1:
Find: When we graph this function, the limit appears to be zero. so for : by the sandwich theorem: Example 2:
Example 3: Find:
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.
Example 4: The denominator is positive in both cases, so the limit is the same.
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 End Behavior Models: End behavior models model the behavior of a function as x approaches infinity or negative infinity.
As , approaches zero. becomes a right-end behavior model. As , increases faster than x decreases, therefore is dominant. becomes a left-end behavior model. Example 7: (The x term dominates.) Test of model Our model is correct. Test of model Our model is correct.
becomes a right-end behavior model. On your calculator, graph: Use: becomes a left-end behavior model. Example 7:
Example 7: Right-end behavior models give us: dominant terms in numerator and denominator
