<s>[INST] What is the $n$ times iterate of $f(x)=\frac{x}{\sqrt{1+x^2}}$? We were asked to determine  the composition $f \circ f \circ f \circ \cdots \circ f $,  $n$ times, where $f(x)=\dfrac{x}{\sqrt{1+x^2}}$.
Has anyone an idea?
 [/INST] Well, try to find some pattern:
$$\begin{align*}f^2(x)=f(f(x))&=f\left(\frac x{\sqrt{1+x^2}}\right)=\frac{\frac x{\sqrt{1+x^2}}}{\sqrt{1+\frac{x^2}{1+x^2}}}=\frac x{\sqrt{1+2x^2}}\\
f^3(x)=f(f^2(x))&=f\left(\frac x{\sqrt{1+2x^2}}\right)=\frac{\frac x{\sqrt{1+2x^2}}}{\sqrt{1+\frac{x^2}{1+2x^2}}}=\frac x{\sqrt{1+3x^2}}\ldots\ldots\end{align*}$$
Now a little induction could help here...
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