multivariable calculus - Limit of $frac{sin(xy^2)}{xy}$

I found this interesting problem on calculating the limit of $\frac{\sin(xy^2)}{xy}$ on the positive coordinate axes $x$ and $y$. That is, compute the limit on the points $(x_0, 0)$ and $(0,y_0)$ when $x_0 > 0$ and $y_0 > 0$.

My approach was this:

If we first calculate the limit for $x$ axis, the the $x$ is a constant $x=x_0$ and therefore the function is essentially a function of one variable:

$$f(y) = \frac{\sin(x_0y^2)}{x_0y}$$

Using L'Hospital's rule:

$$\lim_{y\to0}f(y)=\frac{\lim_{y\to0}\frac{d\sin(x_0y^2)}{dy}}{\lim_{y\to0}\frac{dx_0y}{dy}}$$

$$=\frac{\lim_{y\to0}2yx_0\cos(x_0y^2)}{\lim_{y\to0}x_0}=0$$

The same idea applies to the other limit.

But in the sheet where this problem was listed, this was listed as a "tricky" limit to calculate. It seemed quite simple, so I would like to hear whether this approach is correct. Thank you!

Answer

It is correct, the limit is $0$ and the reasoning holds.