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Evaluate the following limits.

  1. Factor the numerator.


    A limit exists at a hole. What is the -value of the hole?

  1. Notice that this is a limit . Compare the greatest powers of the numerator and denominator.

  1. This is another limit . Compare the terms with the highest powers in the numerator and denominator.

  1. Think about the graph of . It oscillates as . This means that it does not approaches a finite value AND does not approach . When a function oscillates, we say the limit does not exist.

  1. Consider the numerator and the denominator separately.

    As oscillates between and .
    As   approaches .
    So this ratio has small values on the top and infinity on the bottom.