How to compute limit (or show it doesn't exist)?

Question

I've found multiple paths that lead to the following limit being 0, however I can't figure out how to prove this. Thanks

$$\lim_{(x,y)\rightarrow(0,0)}\frac{x^3\cos y}{x^2 +|y|}$$

Answer 1

To prove this, we need to show (from the definition of a limit in two variables) that:

$∀ϵ>0$ $∃δ>0$ such that if $0 < \sqrt{x^2+y^2} < δ$ then $∣f(x,y)∣<ϵ$

I'm going to choose δ=ϵ

If x=0, then the expression evaluates to 0 (except of course at y=0, where it's undefined, but that's the point we're trying to assign a value to in the first place)

Otherwise,

$∣f(x,y)∣=|x^3cos(y)/(x^2+|y|)|\leq|x^3/(x^2+|y|)|\leq|x^3/x^2|=|x|=\sqrt{x^2}\leq\sqrt{x^2+y^2}<δ=ϵ$

QED

Answer 2

In $$x\cos y\frac{1}{1+\dfrac{|y|}{x^2}},$$

the fraction is bounded by $1$ and the the whole expression tends to $0$.