# Product Rule with limits

• Oct 18th 2010, 04:30 PM
zebra2147
Product Rule with limits
Let $\displaystyle n\in N(natural)$ and $\displaystyle a\in C(complex)$. Show that $\displaystyle x^n \rightarrow a^n$ as $\displaystyle a \rightarrow n$. [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.
• Oct 19th 2010, 07:06 AM
Mauritzvdworm
What about x, is there not more structure?

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

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

then
$\displaystyle x^{k+1}=x^kx\rightarrow a^kx$
however the induction hypothesis holds for all $\displaystyle k\geq 1$, so we have
$\displaystyle x^{k+1}\rightarrow a^{k+1}$
• Oct 19th 2010, 07:29 AM
zebra2147
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.