If-f-1-1-and-f-x-1-x-2-f-x-2-then-lim-x-f-x-

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Question Number 264 by novrya last updated on 17/Dec/14

$$Iff\left(1\right)=1andf{}^{\prime }\left(x\right)=\frac{1}{x^2+{\left(f\left(x\right)\right)}^2}then$$$$li\underset{x\to \mathrm{\infty }}{m}f\left(x\right)=\dots .$$

Commented by 123456 last updated on 17/Dec/14

$$\frac{dy}{dx}=\frac{1}{x^2+y^2}$$$$-dx+(x^2+y^2)dy=0,y\left(1\right)=1$$$$\mathrm{M}=-1\Rightarrow {\mathrm{M}}_y=0$$$$\mathrm{N}=x^2+y^2\Rightarrow {\mathrm{N}}_x=2x$$$$-\mu dx+\mu (x^2+y^2)dy=0$$$$\mathrm{M}=-\mu \Rightarrow {\mathrm{M}}_y=-{\mu }_y$$$$\mathrm{N}=\mu (x^2+y^2)\Rightarrow {\mathrm{N}}_x={\mu }_x(x^2+y^2)+2x\mu$$

Commented by 123456 last updated on 21/Dec/14

$$\mathrm{the}\mathrm{equation}\mathrm{is}\mathrm{hard}\mathrm{and}\mathrm{its}\mathrm{possible}\mathrm{that}\mathrm{cannot}\mathrm{be}\mathrm{doing}\mathrm{using}\mathrm{elementary}\mathrm{functions}\dots$$$$\mathrm{i}\mathrm{thinks}\mathrm{thats}\mathrm{exist}\mathrm{other}\mathrm{ways}\mathrm{to}\mathrm{compute}\mathrm{it}\dots$$

Commented by 123456 last updated on 17/Feb/15

$$\underset{1}{\stackrel{x}{\int }}{f}^{\prime }\left(t\right)dt=\underset{1}{\stackrel{x}{\int }}\frac{dt}{t^2+{\left[f\left(t\right)\right]}^2}⩽\underset{1}{\stackrel{x}{\int }}\frac{dt}{t^2+1}$$$$f\left(x\right)-1⩽\frac{\pi }{2}-\frac{\pi }{4}=\frac{\pi }{4}$$$$f\left(x\right)⩽1+\frac{\pi }{4}$$$$f^{\prime }=\frac{1}{x^2+f^2}>0,x⩾1$$$$\frac{1}{x^2+f^2}\to 0,x\to +\mathrm{\infty }$$

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