34. Prove that
\[
\begin{aligned}
\lim _{(x, y) \rightarrow(a, b)}[f(x, y)+g(x, y)] & \\
& =\lim _{(x, y) \rightarrow(a, b)} f(x, y)+\lim _{(x, y) \rightarrow(a, b)} g(x, y)
\end{aligned}
\]
provided that the latter two limits exist.
35. Show that
\[
\lim _{(x, y) \rightarrow(0,9)} \frac{x y}{x^{2}+y^{2}}
\]
does not exist by considering one path to the origin along the \( x \)-axis and another path along the line \( y=x \).
36. Show that
\[
\lim _{(x, y) \rightarrow(0,0)} \frac{x y+y^{3}}{x^{2}+y^{2}}
\]
does not exist.
37. Let \( f(x, y)=x^{2} y /\left(x^{4}+y^{2}\right) \).
(a) Show that \( f(x, y) \rightarrow 0 \) as \( (x, y) \rightarrow(0,0) \) along any straight line \( y=m x \).
(b) Show that \( f(x, y) \rightarrow \frac{1}{2} \) as \( (x, y) \rightarrow(0,0) \) along the parabola \( y=x^{2} \).
(c) What conclusion do you draw?
42. Let \( f(x, y)=x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}} \quad \) if \( \quad(x, y) \neq(0,0) \quad \) and \( f(0,0)=0 \).
Show that \( f_{x y}(0,0) \neq f_{y x}(0,0) \) by completing the following steps:
(a) Show that \( f_{x}(0, y)=\lim _{h \rightarrow 0} \frac{f(0+h, y)-f(0, y)}{h}=-y \) for all \( y \).
(b) Similarly, show that \( f_{y}(x, 0)=x \) for all \( x \).
(c) Show that \( f_{y a}(0,0)=\lim _{h \rightarrow 0} \frac{f_{y}(0+h, 0)-f_{y}(0,0)}{h}=1 \).
(d) Similarly, show that \( f_{x y}(0,0)=-1 \).