1.1.5 Limits At Infinity And Infinite Limits
| Calculus Of One Real Variable – By Pheng Kim Ving |
| 1.1.5 |
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The formal definitions of limits at infinity are stated as follows:
Example 1.1
Find this limit:
Solution
EOS
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| Fig. 2.1y = L is a horizontal asymptote of f. |
Example 2.1
Find horizontal asymptotes of f(x) = 1/x.
Solution
EOS
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As examples we have:
| Fig. 3.1x = a is a vertical asymptote of f. |
Some other examples:
are infinite limits. An infinite limit may be produced by having the independent variable approach a finite point or infinity.
| Note this distinction: a limit at infinity is one where the variable approaches infinity or negative infinity, while an infinite |
The formal definitions of infinite limits are stated as follows:
Example 3.1
Find this limit:
Solution
EOS
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If:
respectively. See Fig. 3.1. The vertical line x = a is called a vertical asymptote of f. Remark that the line x = a can
be a vertical asymptote of f only if f isn’t defined at the point x = a.
Example 4.1
Find vertical asymptotes of f(x) = 1/x.
Solution
EOS
1. Find each of the following limits if it exists. Specify any horizontal or vertical asymptotes of the graphs of the functions.
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2. Find the following limit if it exists. Specify any horizontal or vertical asymptotes of the graph of the function.
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3. Let:
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4. Let:
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5. Find any horizontal and vertical asymptotes of each of the following functions.
Solution
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