$\displaystyle \lim_{x\to a}x^5=a^5 $

I understand that the power rule for limits could be used here but my teacher wants it done using the epsilon delta definition.

1st attempt (though I think it's obviously wrong since my $\displaystyle \delta$ could be undefined):

$\displaystyle |x-a|< \delta \rightarrow |x^5-a^5|< \epsilon$

$\displaystyle |x^5-a^5|=|(x-a)(x^4+ax^3+a^2x^2+a^3x^3 +a^4)|< \epsilon$

$\displaystyle |x-a|< \frac{\epsilon}{|(x^4+ax^3+a^2x^2+a^3x^3 +a^4)|}$

$\displaystyle Choose: \delta = \frac{\epsilon}{|(x^4+ax^3+a^2x^2+a^3x^3 +a^4)|}$