Prove limit of x^5 = a^5 using epsilon delta

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December 14th 2011, 10:17 PM
lamp23

Prove limit of x^5 = a^5 using epsilon delta

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

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 $\delta$ could be undefined):

$|x-a|< \delta \rightarrow |x^5-a^5|< \epsilon$
$|x^5-a^5|=|(x-a)(x^4+ax^3+a^2x^2+a^3x^3 +a^4)|< \epsilon$
$|x-a|< \frac{\epsilon}{|(x^4+ax^3+a^2x^2+a^3x^3 +a^4)|}$

$Choose: \delta = \frac{\epsilon}{|(x^4+ax^3+a^2x^2+a^3x^3 +a^4)|}$
December 14th 2011, 10:26 PM
Chris11

Re: Prove limit of x^5 = a^5 using epsilon delta

You're right that you need to factor. But, you shouldn't have x terms in your choice of delta. Try getting a bound on the big factor (x^4+...) by making a bound on x.
December 14th 2011, 10:47 PM
lamp23

Re: Prove limit of x^5 = a^5 using epsilon delta

Do you mean something very similar to the below example?
Are you suggesting the bound I make is something like $|x-a|<1$ ?
which implies $-1<x-a<1$

$a-1<x<a+1$

$x^4<(a+1)^4$

$|x^3|<|(a+1)^3|$

http://oi39.tinypic.com/ambd6c.jpg
December 15th 2011, 12:36 PM
Chris11

Re: Prove limit of x^5 = a^5 using epsilon delta

Indeed. But don't forget that you want an upper bound, and you certainly have that x< |a|+1 So, use this to get aqn upper bound on the big polynomial factor, take delta to be the minimum of 1 and epsilon/(bound on poly)

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