Chapter 1: Limits
Section 1.5: Limits at Infinity and Infinite Limits
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Example 1.5.4
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Evaluate , where .
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Solution
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Figure 1.5.4(a) has a graph of in red, and graphs of in blue and green, respectively. The blue and green curves are envelopes for the decreasing oscillations of . The figure suggests that .
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Unfortunately, since does not exist, the Product rule for limits cannot be applied. Indeed, since , Principle 1.1.1 could be invoked. Because , it immediately follows that .
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g1 := (x-1)/(x^2+x+1):
f1 := sin(x)*g1:
plot([f1,abs(g1),-abs(g1)],x=10..200,color=[red,blue,green]);
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Figure 1.5.4(a) Graph of
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Application of Maple's limit operator
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Expression palette: Limit operator
A space or explicit multiplication between factors in the numerator is essential.
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Context Panel: Evaluate and Display Inline
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=
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Application of the Squeeze theorem
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With , and because , it follows that
or
Since , both sides of the inequality go to zero, so the middle term must also go to zero, by the Squeeze theorem.
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