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  1. #1
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    example

    Consider a function f: R-------R
    Suppose f is increasing. Is it necessary that that f must approach to infinity?
    For example if f=1/x then it is decreasing and approaches to 0 in (0, infinity) .but that that is possible only when domain is restricted. Is it possible on R for some function which are either increasing or decreasing on R but does not approach to +infinity or -infinity?
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  2. #2
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    Monotonic function - Wikipedia, the free encyclopedia

    See "Some basic applications and results".
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  3. #3
    Behold, the power of SARDINES!
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    Quote Originally Posted by ayushdadhwal View Post
    Consider a function f: R-------R
    Suppose f is increasing. Is it necessary that that f must approach to infinity?
    For example if f=1/x then it is decreasing and approaches to 0 in (0, infinity) .but that that is possible only when domain is restricted. Is it possible on R for some function which are either increasing or decreasing on R but does not approach to +infinity or -infinity?
    Consider the function f(x)=\tan^{-1}(x)
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  4. #4
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    We can take 1/x when x >= 1, reflect it with respect to x-axis to get -1/x, shift it left by 1 and up by 1 to get g(x) = -1/(x + 1) + 1. Here is its graph plotted by WolframAlpha. Then define f(x)=<br />
\begin{cases}<br />
g(x), & x \ge 0\\<br />
-g(-x), & x < 0<br />
\end{cases}. I.e., to get the graph of f(x) for x < 0, we rotate the graph of g(x) for x >= 0 by 180 degrees. The result is a smooth function that tends to -1 when x\to-\infty and to 1 when x\to\infty.

    For another example, consider arctan(x).
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