2.2 Limits Involving Infinity North Dakota Sunset Photo by

2.2 Limits Involving Infinity North Dakota Sunset Photo by Vickie Kelly, 2006 Greg Kelly, Hanford High School, Richland, Washington

1 f x x 1 lim 0 x x As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if: lim f x b x or lim f x b x

Example 1: lim x x x2 1 lim x x x2 x lim 1 x x This number becomes insignificant as x . There is a horizontal asymptote at 1.

sin x f x x Example 2: When we graph this function, the limit appears to be zero. 1 sin x 1 so for x 0 : Find: sin x lim x x 1 sin x 1 x x x 1 sin x 1 lim lim lim x x x x x x sin x 0 lim 0 x x by the sandwich theorem: sin x lim 0 x x

Example 3: Find: 5 x sin x lim x x 5 x sin x lim x x x sin x lim 5 lim x x x 5 0 5

Infinite Limits: 1 f x x 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. 1 lim x 0 x 1 lim x 0 x

Example 4: 1 lim 2 x 0 x 1 lim 2 x 0 x The denominator is positive in both cases, so the limit is the same. 1 lim 2 x 0 x

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 if f x lim 1 x g x a left end behavior model for f if and only if f x lim 1 x g x

Example 7: f x x e x x x e As , approaches zero. (The x term dominates.) g x x becomes a right-end behavior model. Test of model x e lim x x x e x 1 0 1 lim1 x x Our model is correct. x As x , e increases faster than x decreases, x therefore e is dominant. h x e x becomes a left-end behavior model. Test of model x x e x lim 1 0 1 1 lim x x x x e e Our model is correct.

Example 7: f x x e x g x x becomes a right-end behavior model. h x e x becomes a left-end behavior model. On your calculator, graph: y1 x y2 e x y3 x e x Use: 10 x 10 1 y 9

Example 7: 2 x5 x 4 x 2 1 f x 3x 2 5 x 7 Right-end behavior models give us: 2 x5 2 x3 2 3x 3 dominant terms in numerator and denominator

Often you can just “think through” limits. 1 lim sin x x 0 lim sin x x 0 0