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Limits Involving Infinity

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  1. Photo by Vickie Kelly, 2006 Greg Kelly, Richland, Washington Limits Involving Infinity North Dakota Sunset

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

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

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

  5. Example 3: Find:

  6. 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.

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

  8. 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.

  9. 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.

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

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

  12. Often you can just “think through” limits.