example

• April 10th 2011, 12:03 PM
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?
• April 10th 2011, 12:10 PM
veileen
Monotonic function - Wikipedia, the free encyclopedia

See "Some basic applications and results".
• April 10th 2011, 12:29 PM
TheEmptySet
Quote:

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 $f(x)=\tan^{-1}(x)$
• April 10th 2011, 12:30 PM
emakarov
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 $f(x)=
\begin{cases}
g(x), & x \ge 0\\
-g(-x), & x < 0
\end{cases}$
. 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).