Prove that $\lim_{t\to 0}f(g(t))$ exists, where $g$ is differentiable and $f(x,y)=x^2y/(x^2+y^2)$.

Prove that $\lim_{t\to 0}f(g(t))$ exists, where $g$ is differentiable and
$f(x,y)=x^2y/(x^2+y^2)$.

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a function defined by
$$f(x,y)=\left\{\begin{matrix} \frac{x^2y}{x^2+y^2}&\text{if
}(x,y)\neq(0,0) \\ 0&\text{if }x=y=0 \end{matrix}\right.$$
Let $\varepsilon>0$ and $g:(-\varepsilon,\varepsilon)\to\mathbb{R}^2$ be a
function differentiable at point $0$ such that $g(0)=(0,0)$. Prove that
the following limit exists.
$$\lim_{t\to 0}\frac{f\left(g(t)\right)}{t}$$
Thanks.