TITLE: How to use inequalities to solve for multivariable limits?
QUESTION [5 upvotes]: I am searching for a  methodological description on how to select the correct functions over several inequalities to find and prove that a function has a limit. I quote an example
Find the limit of the given function:
\begin{equation}
\lim_{(x,y)\longrightarrow(0,0)}\frac{x^2y}{x^2+y^2}
\end{equation}
The solution, in terms of inequalities , is:
\begin{equation}
\bigg|\frac{x^2y}{x^2+y^2}\bigg|\leqslant|y|\leqslant\sqrt{x^2+y^2}
\end{equation}
where $\sqrt{x^2+y^2}$ approaches $0$ as $(x,y)\longrightarrow(0,0)$.
First of all, where does the author of this solution get those functions between the intervals, in the solution? What rationale does he use, when he constructs these functions?
Apparently , there are many different functions that have limits that reach zero, so is this selection purely intuitive, and a subjective choice? If so, what is the objective method behind this?
Thanks

REPLY [2 votes]: There is not an unique general way to solve this kind of limits, the way proposed is indeed a valid one among others.
First of all, assuming that the limit could be equal to zero, it is convenient to consider the absolute value for the given expression, indeed
$$|f(x)| \to 0 \implies f(x) \to 0$$
to obtain
$$\left|\frac{x^2y}{x^2+y^2}\right|=\frac{x^2|y|}{x^2+y^2}$$
then, in order to use inequalities we can observe that
$$\frac{x^2|y|}{x^2+y^2} \le \frac{x^2|y|+y^2|y|}{x^2+y^2}=|y| \to 0$$
and conclude.
The way I prefer in this case is by polar coordinates, to obtain:
$$\frac{x^2y}{x^2+y^2} = \rho \cos^2 \theta \sin \theta \to 0$$
which is simpler but, in general, needs to be used with caution. On that last issue, refer to the related:

Why is this method wrong on calculating this multivariable limit?