We were given the following function:
$$Q(h)\equiv\frac{f(x+h,y)+f(x-h,y)+f(x,y+h)+f(x,y-h)-4f(x,y)}{h^2}$$
which is the approximation of the laplacian of $f$ at $x$ and $y$ and asked to find the limit of $Q(h)$ as $h\to0$ in terms of the partial derivatives of $f$.
I know that the definition for partial derivatives is
$$f_x(x,y)=\frac{f(x+h,y)-f(x,y)}h$ and $f_y(x,y)=\frac{f(x,y+h)-f(x,y)}h$$
and with this I can spread it out to
$\begin{aligned}\lim_{h\to 0}Q(h)&= \lim_{h\to0}\frac1h\lim_{h\to0}\frac{f(x+h,y)-f(x,y)}h+ \lim_{h\to0}\frac1h\lim_{h\to0}\frac{f(x-h,y)-f(x,y}h+ \lim_{h\to0}\frac1h\lim_{h\to0}\frac{f(x,y+h)-f(x,y)}h + \lim_{h\to0}\frac1h\lim_{h\to0}\frac{f(x,y-h)-f(x,y)}h\end{aligned}$
and then, from there get
$\begin{aligned}\lim_{h\to0}Q(h)&= \lim_{h\to0}\frac1hf_x(x,y)+\lim_{h\to0}\frac1h\lim_{h\to 0}\frac{f(x-h,y)-f(x,y)}h+ \lim_{h\to0}\frac1hf_y(x,y) + \lim_{h\to0}\frac1h\lim_{h\to0}\frac{f(x,y-h)-f(x,y)}h,\end{aligned}$
but I get stuck with this: $\lim\limits_{h\to0}\frac1h$ which is basically infinity and also $\lim\limits_{h\to0}\frac{f(x-h,y)-f(x,y)}h$ and $\lim\limits_{h\to0}\frac{f(x,y-h)-f(x,y)}h$ don't know what to do with them.
Any help is appreciated.
This should help.
$\frac {\partial f}{\partial x} = \lim_\limits{h\to0} \frac {f(x+h,y) - f(x,y)}{h}\\ \frac {\partial^2 f}{\partial x^2} = \lim_\limits{h\to0} \frac {(f(x+h,y) - f(x,y)) - (f(x-h,y) - f(x))}{h^2}$