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I'm working through an advanced calculus book and want to be certain I understand the idea behind proving limits. This is not homework, I'm just a statistician looking to learn more about mathematics.

The exercise I'm concerned with proving is as follows:

$\begin{aligned} \lim_{(x,y)→(0,0)} \frac{x^3y}{x^2 + y^4} \\\ \end{aligned}$

My understanding is that I can choose a value to substitute in for y that allows for some easy cancellation that proves the limit equals 0. For instance:

$\begin{aligned} x= y^2 \ ; \frac{(y^2)^3y}{(y^2)^2 + y^4} \\\ \end{aligned}$

From here, we have:

$\begin{aligned} \frac{y^7}{y^4(1 + 1)} \\\ \end{aligned}$

Then as y→0 this simplifies to:

$\begin{aligned} \frac{0^3}{2} = 0 \\\ \end{aligned}$

Is this how the limit could/would be proved?

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We observe that $ 0\leq\left|\frac{x^3y}{x^2+y^4}\right|\leq \frac{|x^3y|}{x^2}=|xy| $ for all $x,y\ne 0$. Since $\displaystyle\lim_{(x,y)\rightarrow (0,0)}|xy|=0$ then $ \lim_{(x,y)\rightarrow (0,0)}\frac{x^3y}{x^2+y^4}=0. $

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    @Jim M: You are welcome. I am ready to help you.2012-10-05