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Chapter 1 Limits and Continuity

Outline 7. Infinite limits and Limits at

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7. Infinite Limits Rule provided

and

provided

and

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EX Show that

Remark For infinite limits, the quotient law and prelim. algebra cannot be used.

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EX (One-sided infinite limits) Evaluate the limits

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Limits At Def We say that

if the values can be arbitrarily close to by taking large enough.

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Def We say that

if the values can be arbitrarily close to by taking small enough.

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EX Find the limit

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Def If either

then the horizontal line

is called a horizontal asymptote of the graph

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Remark

1. All limit laws and facts can be applied.

2. We shall use the fact that

and the limit laws

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EX Find the infinite limits, limits at , and the asymptotes for the function whose graph is shown below.

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EX (Divide by highest power of ) Evaluate

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EX (Do rationalization first) Evaluate

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EX ( ) Evaluate

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Def We say that

if the values of can be arbitrarily large (resp. small) as is large enough. We say that

if the values of can be arbitrarily large (resp. small) as is small enough.

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EX Find the limit

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EX Find

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EX Find