<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] Let's denote $f^n = f \circ ... \circ f$ with $n$ times the function $f$, and suppose (which is obviously true for $n=1$), that:
$$f^n(x) = \frac{x}{\sqrt{1+n x^2}}$$
Then
$$f^{n+1}(x)=f \circ f^n(x)= \frac{\frac{x}{\sqrt{1+n x^2}}}{\sqrt{1+\left(\frac{x}{\sqrt{1+n x^2}}\right)^2}}$$
$$=\frac{\frac{x}{\sqrt{1+n x^2}}}{\sqrt{\frac{1+(n+1)x^2}{1+nx^2}}}=\frac{x}{\sqrt{1+(n+1)x^2}}$$
And you are done, by induction.
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