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Limits at Infinity (End Behavior) Section 2.3

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Some Basic Limits to Know Lets look at the graph of What happens at x approaches -? What happens as x approaches ? On the original graph, where is the horizontal asymptote? General rule: – If, then there is a horizontal asymptote at y = L. (Think about why this is.)

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Limits Using Degrees Lets look at pg 125. Pay close attention to the blue boxes! *Big fact: – Multiplying x n by a negative number will reverse the sign as well as the end behavior! Ex:

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Limits Using Degrees These tricks also apply to polynomials! Just look at the degree to find the limit. Ex:

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Rational Functions Long way: Divide each term by the highest power of x in the denominator. Ex: – Divide everything by x. – Simplify: – Use the limit just learned!

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Rational Functions Lets try this one on your own…. Ex: Answer should be … 0

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Rational Functions One more… Ex: Oh no! What is the issue? Recall the 4 steps for limits… So:

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Rational Functions Fast way: Use degrees to find end behavior! Use the term with the highest power in numerator and denominator to decide behavior. Ex:

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Limits With Radicals Same idea, just watch the radical signs! Ex: Only change here is the answer is under the radical:

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Limits With Radicals Gets harder when only one side of the fraction is under a radical sign. Ex:Still divide everything by x! Why are we using x 2 in the numerator Lets finish:

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Limits With Radicals New twist… Ex:What is different? Now we need to divide by -. Why? Again, why x 2 in the radical? Lets finish:

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Limits With Radicals Lets really have a challenge! Ex: Remind you of something? Lets make this a rational function: Lets finish:

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Examples for Rationalizing the Denominator Examples It is improper for a fraction to have a radical in its denominator. To remove the radical we “rationalize.

Examples for Rationalizing the Denominator Examples It is improper for a fraction to have a radical in its denominator. To remove the radical we “rationalize.

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