As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if: or

Example 1: This number becomes insignificant as. There is a horizontal asymptote at 1.

Example 2: Find: When we graph this function, the limit appears to be zero. so for : by the sandwich theorem:

Example 3: Find:

Infinite Limits: As the denominator approaches zero, the value of the fraction gets very large. If the denominator is positive then the fraction is positive. If the denominator is negative then the fraction is negative. vertical asymptote at x =0.

Example 4: The denominator is positive in both cases, so the limit is the same.

End Behavior Models: End behavior models model the behavior of a function as x approaches infinity or negative infinity. A function g is: a right end behavior model for f if and only ifa left end behavior model for f if and only if

Test of model Our model is correct. Example 7: As, approaches zero. (The x term dominates.) becomes a right-end behavior model.becomes a left-end behavior model. As, increases faster than x decreases, therefore is dominant. Test of model Our model is correct.

Example 7: becomes a right-end behavior model.becomes a left-end behavior model. On your calculator, graph: Use:

Example 7: Right-end behavior models give us: dominant terms in numerator and denominator

Often you can just “think through” limits. 