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