Limit of function as $x \to\infty $ when $f'(x)$ is given

Question

Let $f:[1,\infty)\rightarrow \mathbb R $ is a differentiable function which satisfies $$f'(x)=\frac {1}{x^2+(f (x))^2} \text{ and } f(1)=1$$ then find the limit of $f $ as $x \to\infty $

My attempt: So I first thought of making a differential equation and then calculate the limit. But the differential equation formed $y'(y^2+x^2)=1$ is a non standard equation and it cannot be solved. Even calculators on the internet show that "no solution found". However, Wolfram Alpha does provide a graph but not the solution.

Also $f^{\prime\prime}(x)<0$. Can this fact be used in some way?

Next I thought of using Rolle's theorem but I am unable to figure out some way to use it.

Can anyone provide me some idea on how to approach this problem?

Answer 1

The inverse function satisfies $$x'(f)=x(f)^2+f^2$$ which is a classical Riccati equation. The pole of its solution will be the limit that you are looking for.

You can find a solution in terms of Bessel functions:

• Riccati differential equation $y'=x^2+y^2$

You can try various estimates with methods like in

• Riccati D.E., vertical asymptotes
• $ y' = x^2 + y^2 $ asymptote

or you can compute a numerical solution.

Answer 2

As $f'\left(x\right) >0, f\left(x\right)$ is an increasing function, so we have, for $t>1,f\left(x\right)>f\left(1\right)=1$, Therefore, for $ t>1$, \begin{align} f\left(x\right)&=1+\int_{1}^xf'\left(t\right) dt \\ &<1+\int_{1}^x\frac{1}{t^2+1} dt \\ &<1+\int_{1}^{\infty}\frac{1}{t^2+1} dt \\ &=1+\frac{\pi}{4} ; \end{align} hence, $\lim\limits_{x\to\infty}f\left(x\right)$ exists and is, at most, $$1+\frac{\pi}{4}$$