For a math course, my course book computes the limit of the function $f(x,y) = \dfrac{xy^2}{x^2+y^4}$ at $(0,0)$ along the $y$-axis (along $\mathbf{r}(t) = (0,t)$). It finds $\lim _{t \rightarrow 0}f(\mathbf{r}(t)) = \lim _{t \rightarrow 0}\dfrac{0}{0+t^4} = 0\\$.
My question is why is this limit $0$ when plugging $t=0$ in will give $\dfrac{0}{0}$. I have tried to find similar problems online to compare but every time I find a similar problem it provides the answer $0$ without a good explanation.
We are not plugging in values just yet. First we simplify, as $t\neq0$: $$ \lim_{t\to 0}\frac0{0+t^4}=\lim_{t\to0}0 $$ Now we can plug in $0$ for $t$ in the right-hand side and evaluate the limit. We see that it is $0$.