(a) Let \( x=\alpha(t), y=\beta(t) \) be the parametric equations of a path in \( \mathbb{R}^{2} \) such that \( (\alpha(t), \beta(t)) \) lies in the domain of \( f(x, y) \), for all \( t \), in a certain open interval containing \( t_{0} \) and
\[
\lim _{t \rightarrow t_{0}} \alpha(t)=a \text { and } \lim _{t \rightarrow t_{0}} \beta(t)=b .
\]
Prove that if \( \lim _{(x, y) \rightarrow(a, b)} f(x, y)=L \) then \( \lim _{t \rightarrow t_{0}} f(\alpha(t), \beta(t))=L \).
Hence show that the following limits do not exist:
i. \( \lim _{(x, y) \rightarrow(1,0)} \frac{y}{x+y-1} \),
ii. \( \lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}+3 x^{2} y^{2}+2 x y^{3}}{\left(x^{2}+y^{2}\right)^{2}} \).