<s>[INST] Limit of the difference quotient of $f(x) = \frac{2}{x^2}$, as $x\rightarrow x_0$ Could someone please show me how to derive the limit of the difference quotient of $f(x) = \frac{2}{x^2}$, as $x\rightarrow x_0$
The difference quotient is just the expression: $(f(x+h)-f(x))/h$
So, it's easy to get an expression for this, but a little tricky to eliminate the 'h' from the denominator.. 
Would appreciate it if someone can show me how. 
 [/INST] The trick is simply to add the two fractions together.
$$\lim_{h \to 0} \frac{\frac{2}{(x + h)^2} - \frac{2}{x^2}}{h} = \lim_{h \to 0} \frac{\frac{2[x^2 - (x+h)^2]}{x^2(x + h)^2}}{h} = \lim_{h \to 0} \frac{\frac{2[-2hx - h^2]}{x^2(x + h)^2}}{h} = \lim_{h \to 0} \frac{2[-2x - h]}{x^2(x + h)^2} = \frac{2(-2x)}{x^4} = \frac{-4}{x^3}$$
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