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Uni Stuff => Science => Faculties => Mathematics => Topic started by: Cthulhu on March 22, 2010, 12:20:38 am

Title: Multivariable Limits
Post by: Cthulhu on March 22, 2010, 12:20:38 am: HEY THAR MATHEMATICIANS.
I've gone over this 100 times already and I cant seem to do it.
I need to the use $\epsilon,\delta$ definition of a limit to show that something like (not the exact question I just want to see how I'd do something like this.)
$ \lim_{(x,y) \to (0,0)} \frac{xy^5}{x^2 + y^2} = 0 $
I absolutely fail at epsilon delta shit.
Your help is appreciated.
Title: Re: Multivariable Limits
Post by: kamil9876 on March 22, 2010, 06:03:29 pm: let $f(x)=\frac{xy^5}{x^2 + y^2} $

let $r=\sqrt{x^2+y^2}$

we immediately get: $|x| \leq r$ and $|y| \leq r$

$ \therefore |f(x)|<\frac{r^6}{r^2}=r^4$

So it really just reduces to showing that $r^4$ can be made arbitrarily small for small enough $r$ .

If you wanna make $|f(x)|<\epsilon$ for $\epsilon\geq 1$ just set $\delta=1$ since for $r<1 \implies |f(x)|<r^4<1\leq \epsilon$

Whereas if you wanna make $|f(x)|<\epsilon$ for $0<\epsilon<1$ just set $\delta=\epsilon$ since $r<\epsilon \implies |f(x)|<r^4<\epsilon^4<\epsilon$

$\epsilon-\delta$ Exercise: Prove that the part $\epsilon \geq 1$ in red is redundant.
Title: Re: Multivariable Limits
Post by: QuantumJG on March 22, 2010, 06:23:41 pm: Quote from: Cthulhu on March 22, 2010, 12:20:38 am
HEY THAR MATHEMATICIANS.
I've gone over this 100 times already and I cant seem to do it.
I need to the use $\epsilon,\delta$ definition of a limit to show that something like (not the exact question I just want to see how I'd do something like this.)
$ \lim_{(x,y) \to (0,0)} \frac{xy^5}{x^2 + y^2} = 0 $
I absolutely fail at epsilon delta shit.
Your help is appreciated.

Lol we are doing ε-δ proofs now... but for one variable.

How is quantum mechanics going?