Question

lim h -> 0 ((1/(x+h)^2) - 1/x^2)/h

It's a bit messy, but can anyone help?

Answer 1

\[\frac{ x^{2} - (x+h)^{2} }{ x^{2}h(x+h)^{2} }= \frac{ x^{2}-(x^{2}+2xh+h^{2}) }{ x^{2}h(x+h)^{2} }\]

Simplify top.

\[\frac{ x^{2}-x^{2}-2xh-h^{2} }{ x^{2}h(x+h)^{2} }\]=
\[\frac{ -2xh-h^{2} }{ x^{2}h(x+h)^{2} }\]
Factor out an h from the top and let it cancel out with the h on bottom:

\[\frac{ h(-2x-h) }{ x^{2}h(x+h)^{2} }= \]=\[\frac{ -2x-h }{ x^{2}(x+h)^{2} }\]
Let h go to 0
\[\frac{ -2x-0 }{ x^{2}(x+0)^{2} }= \frac{ -2x }{ x^{4} }= -\frac{ 2 }{ x^{3} }\]