Let f:(0,)(0,) be differentiable, f(1)=e, and _t xt^2f^2(x)-x^2f^2(t)t-x=0.

Question

Let $f:(0,\infty)	o(0,\infty)$ be differentiable, $f(1)=e$, and $\lim_{t	o x}\dfrac{t^2f^2(x)-x^2f^2(t)}{t-x}=0$. If $f(x)=1$, then $x$ equals:

Accepted Answer

By L'Hopital, $f(x)=xf'(x) \implies f(x)=Cx$; using $f(1)=e \Rightarrow C=e$, so $f(x)=ex$.

$f(x)=1 \implies x=1/e$.

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