Uniform limit depending of analytique functions

Question

can we say that $\exp(-1/x^n)$ converge uniformly to $1-1_{[0,1]}$?

what can we say, if we put $f$ a (real) entire function, of the limit : $$ \lim_{n\to\infty} \frac{f(0)}{f(1/x^n)} $$

Answer 1

An elementary result is that the uniform limit of continuous functions on an interval has a continuous limit. Restricting your sequence to, say, $[-2,2]$, then if the sequence converged uniformly, then the limit would necessarily be continuous, which it is not.

Answer 2

No on the first question, because $f_n(1) = 1/e$ for all $n,$ but $1-1_{[0,1]} =0$ at $1.$