Anyone can help me ?
QUESTION :
for all x in (0,1), n is Natural number
f_n (x) := max{1/x , n }
find lim f_n (x) when n -> infinity.
Do you mean {1/n, n}? If so then you are looking at "max(1/2, 2)", "max(1/3, 3)", "max(1/4, 4)"...:"max{1/1000000, 1000000}", ... What are those numbers? What is the limit as n goes to infinity?
If you really mean {1/x, n} for fixed x it is essentially the same: "max(1/x, 2)", "max(1/x, 3)", "max(1/x 4)"...:"max{1/x 1000000}", where the first number is some fixed finite number.
It's pretty straight forward if you write out a few terms in the sequence of functions.
For x in (0, 1) (in particular for x positive) 1/x> n when x< 1/n. So $\displaystyle f_n(x)= 1/x$ for x< 1/n and $\displaystyle f_n(x)= n$ for x> 1/n, In particular, $\displaystyle f_1(x)= 1/x$ for all x in (0, 1). $\displaystyle f_2(x)= 1/x$ for 0< x<= 1/2, $\displaystyle f_2(x)= 2$ for 1/2<= x< 1. $\displaystyle f_3(x)= 1/x$ for 0< x<= 1/3 and $\displaystyle f_3(x)= 3$ for 1/3<= x< 1. Do see what is happening? Given any x, there is eventually an integer N> 1/x so for n> N, $\displaystyle f_n(x)> N$.