## oksuz_ one year ago help!!

1. oksuz_

show that the sequence of function given by $fn(x)=nlog(1+\sin(x/n))$ converges point wise to the identity function f(x)=x anyone has an idea ?

2. BSwan

like $$f_1(x)= log (1+sin x )$$ $$f_2(x)=f_1+2 log (1+sin x/2 )$$ ... ?!

3. klimenkov

Use this: $$\sin x = x + O(x^3), \, x\rightarrow 0$$. $\lim_{n\rightarrow \infty}n\log\left(1 + \sin\left( x/n\right)\right) = \lim_{n\rightarrow \infty}\log\left(1 + x/n\right)^n$

4. BSwan

it would be something like this :- |dw:1401617819184:dw|

5. oksuz_

@klimenkov at your solution last part equals e^x right? then when x goes to infinite e^x also goes to infinite.. should we consider identity function's interval here that is convergent ??

6. klimenkov

$$x$$ is a fixed point and $$n\rightarrow \infty$$. So it will converge for every finite $$x$$. $\log e^x = x.$

7. oksuz_

@klimenkov thank you..

8. oksuz_

@BSwan thank you