# Is $\lim_{n \to \infty} x^n$ continuous on [0,1]?

Is $$\lim_{n \to \infty} x^n$$ a continuous function on [0,1]?

PS: The original question was for $\lim_{n \to \infty} (\sin(x))^n$ but it brought it complications that are not relevant to the main idea.

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This function does not defiened at $x=(2n+3/2)\pi$ for all $n\in\mathbb{Z}$. –  tetori Jan 6 '13 at 7:24
What is the domain over which you are interested in continuity? Is it the entire $\mathbb{R}$? –  user17762 Jan 6 '13 at 7:24
If you add an absolute value sign, the function still remains discontinous –  Amr Jan 6 '13 at 7:25
Arjang: The purpose of this homework is to make you identify the limit pointwise. That is, fix $x$, does the limit exist and what is it? –  Did Jan 6 '13 at 7:27
Thanks for the explanations, unfortunately I fail to see the relevance of this question for the two others, nor the other way round. (If the problem is to find properties which fail to be automatically satisfied when one passes to the limit (whatever that means), try this: every integer $n$ is finite; let $n\to\infty$; then...) Anyway. –  Did Jan 6 '13 at 11:33

What is $\lim_{n \to \infty}1^n$?
What is $\lim_{n \to \infty}a^n$ for $a \in [0,1)$?