Question

help!!

Answer 1

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 ?

Answer 2

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

Answer 3

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

Answer 4

it would be something like this :-