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Given the function , calculate the following limits without using your graphing calculator. 

Compute without a calculator

  1. Evaluate.

  1. Before finding the limit, factor the numerator and the denominator. If the '' in the denominator cancels out, then the limit exists.

  1. For all limits as , you are looking for end behavior. For example, is there a horizontal asymptote? Or does the function approach or in the end? To answer this question, compare the highest power in the numerator with the highest power in the denominator.

    Highest power in the numerator:

    Highest power in the denominator:

    Note: this means that has a horizontal asymptote of .

  1. First, factor the expression. Visualize the left-hand and right-hand limits of the simplified function. Do they match up?

    You can prove this by showing that

    Limit does not exist.