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## Limits Involving Infinity

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**Photo by Vickie Kelly, 2006**Greg Kelly, Richland, Washington 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 squeeze 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