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?