Product Rule with limits

**zebra2147** · October 18th 2010, 04:30 PM

Let $n\in N(natural)$ and $a\in C(complex)$ . Show that $x^n \rightarrow a^n$ as $a \rightarrow n$ . [Hint:use the product rule and induction on n]

I am studying for an exam when i came across this problem in my notes. Any help would be appreciated.

**Mauritzvdworm** · October 19th 2010, 07:06 AM

What about x, is there not more structure?

In order to use induction we will need $x^1\rightarrow a^1$ for the initial step

Then for the induction hypothesis we assume
$x^k\rightarrow a^k$ for all $k\geq 1$

then
$x^{k+1}=x^kx\rightarrow a^kx$
however the induction hypothesis holds for all $k\geq 1$ , so we have
$x^{k+1}\rightarrow a^{k+1}$

**zebra2147** · October 19th 2010, 07:29 AM

No, that's all that we are given with regards to x. But your explanation is very helpful. For some reason I couldn't out the induction process together with the product rule.

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