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Limits involving infinity
Section 2.2 Limits involving infinity
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Finite Limits as x Approaches Infinity
Definition 1: The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either
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Horizontal Asymptotes
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Example 1 Looking for horizontal asymptotes.
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Example 2 Use graphs and tables to find (a) and (b). (c) Identify all horizontal asymptotes.
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Example 3 Use graphs and tables to find (a) and (b). (c) Identify all horizontal asymptotes.
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Example 4 Use graphs and tables to find (a) and (b). (c) Identify all horizontal asymptotes.
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Infinite Limits as x Approaches a
Definition 2: The line x = a is a vertical asymptote of the graph of a function y = f(x) if either
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Vertical Asymptote
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Example 5 (a) Find the vertical asymptotes of the graph of f(x). (b) Describe the behavior of f(x) to the left and right of each vertical asymptote.
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Example 6 (a) Find the vertical asymptotes of the graph of f(x). (b) Describe the behavior of f(x) to the left and right of each vertical asymptote.
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TOTD Use graphs and tables to find limits.
Use graphs and tables to find the limit as x approaches Β±β. Identify any horizontal asymptotes. π¦= π₯ π₯
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Example 7 (a) Find the vertical asymptotes of the graph of f(x). (b) Describe the behavior of f(x) to the left and right of each vertical asymptote.
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Example 8 (a) Find the vertical asymptotes of the graph of f(x). (b) Describe the behavior of f(x) to the left and right of each vertical asymptote.
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Example 9
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Example 10
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TOTD Find the vertical asymptotes of the graph of f(x). Describe the behavior of f(x) to the left and right of each vertical asymptote. π π₯ = π₯ 2 β1 2π₯+4 Use graphs and tables to find the limit as x approaches Β±β. Identify any horizontal asymptotes. π¦= π₯π πππ₯+2π πππ₯ 2 π₯ 2
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Example 11 Show that while f and g are quite different for numerically small values of x, they are virtually identical for |x| large.
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End Behavior Models The function is
(a) a right-end behavior model for iff (b) a left-end behavior model for iff
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Example 12 (a) Find a power function end behavior model for f. (b) Identify any horizontal asymptotes.
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Example 13 (a) Find a power function end behavior model for f. (b) Identify any horizontal asymptotes.
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Example 14 Find (a) a simple basic function as a right end behavior model and (b) a simple basic function as a left end behavior model for the function.
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Example 15 Find (a) a simple basic function as a right end behavior model and (b) a simple basic function as a left end behavior model for the function.
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Example 16
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Example 17
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