Consider the function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ defined by $f(x, y)=x^{2}(y-1)$. For $\vec{u}=\left(\frac{1}{2}, \frac{1}{2}\right)$ and $\vec{v}=(3,4)$, the value of the limit
$$ \lim _{t \rightarrow 0} \frac{f(\vec{u}+t \vec{v})-f(\vec{u})}{t} $$
is
- $\frac{3}{4}$
- $\frac{6}{13}$
- $-\frac{1}{2}$
- none of the above
