Information AboutAsymptote |
| CATEGORIES ABOUT ASYMPTOTE | |
| mathematical analysis | |
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An asymptote is a straight or curved Line which a curve will approach arbitrarily closely, but never touch. A specific example of asymptotes can be found in the asymptote (such as the preceding example's ''x'' = 0, which has an undefined Slope ) could be said to approach an " Infinite Limit ," while a curve approaching a Horizontal line (such as the previous example's ''y'' = 0) could be said to approach a Limit at infinity. Asymptotes need not be parallel to the ''x''- or ''y''-axis, as shown by the graph of ''x'' + ''x''−1, which is asymptotic to both the ''y''-axis and the line ''y'' = ''x''. When an asymptote is not parallel to the ''x''- or ''y''-axis, it is called an oblique asymptote. Asymptotes, especially vertical asymptotes, also do not need to go to infinity when approached at both sides. Asymptote x=a is a vertical asymptote for f(x) if it just satisfies ''either'' of the following conditions: # # Note that ''f''(''x'') need not be undefined at ''a''. For example, consider the function : As both and , ''f''(''x'') has a vertical asymptote at 0, even though . A function ''f''(''x'') can be said to be asymptotic to a function ''g''(''x'') as ''x'' → ∞. This has any of four distinct meanings: # ''f''(''x'') − ''g''(''x'') → 0. # ''f''(''x'') / ''g''(''x'') → 1. # ''f''(''x'') / ''g''(''x'') has a nonzero limit. # ''f''(''x'') / ''g''(''x'') is bounded and does not approach zero. See Big O Notation . See also Asymptotic Analysis , but contrast with Asymptotic Curve |
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