Squeeze Theorem

If $f(\vec{v})=g(\vec{v})h(\vec{v})$ where ${\lim }_{\vec{v}\to {\vec{v}}_0}g(\vec{v})=0$ and $|h(\vec{v})|\le C$, then ${\lim }_{\vec{v}\to {\vec{v}}_0}f(\vec{v})=0$ if you have a product between something whose limit goes to 0, and some other function that is bounded, then the whole thing will go to 0.

Example

$$\underset{(x,y)\to (0,0)}{\lim }\frac{x^{4}y}{x^4+y^2}$$ $$\text{Idea: }\frac{x^{4}y}{x^4+y^2}=\underset{\text{bounded}}{\underbrace{\frac{x^4}{x^4+y^2}}}\cdot \underset{\text{goes to 0}}{\underbrace{y}}$$

In this case the bounded term is leq 1, and y goes to 0 Therefore the whole thing goes to 0, by the Squeeze Theorem, QED.