Answer
a) $8x+4h$
b) $-4y-2h$
c) $8x$
d) $-4y$
Work Step by Step
$a)\frac{f(x+h,y)-f(x,y)}{h}=\frac{4(x+h)^{2}-2y^{2}-(4x^{2}-2y^{2})}{h}=8x+4h$
$b)\frac{f(x,y+h)-f(x,y)}{h}=\frac{4x^{2}-2(y+h)^{2}-(4x^{2}-2y^{2})}{h}=-4y-2h$
$c)\lim\limits_{h \to 0}\frac{f(x+h,y)-f(x,y)}{h}=\lim\limits_{h \to 0}(8x+4h)=8x+4\cdot 0=8x$
$d)\lim\limits_{h \to 0}\frac{f(x,y+h)-f(x,y)}{h}=\lim\limits_{h \to 0}(-4y-2h)=-4y-2\cdot 0=-4y$
