Question

SO A LETRA C AI PESSOAL, A QUE TEM,
f(x)= 1/x+2​

Answer 1

$\underset{\Delta x\to0}{lim}~\frac{\frac{1}{x+\Delta x+2}-\frac{1}{x+2}}{\Delta x}~=~\\\\\\\underset{\Delta x\to0}{lim}~\frac{\frac{1~.~(x+2)~-~1~.~(x+\Delta x+2)}{(x+\Delta x+2).(x+2)}}{\Delta x}\\\\\\\underset{\Delta x\to0}{lim}~\frac{1~.~(x+2)~-~1~.~(x+\Delta x+2)}{(x+\Delta x+2).(x+2).\Delta x}\\\\\\\underset{\Delta x\to0}{lim}~\frac{x+2-x-\Delta x-2}{(x+\Delta x+2).(x+2).\Delta x}\\\\\\\underset{\Delta x\to0}{lim}~\frac{-\Delta x}{(x+\Delta x+2).(x+2).\Delta x}$

$\underset{\Delta x\to0}{lim}~\frac{-1}{(x+\Delta x+2).(x+2).1}~=~\frac{-1}{(x+0+2).(x+2)}\\\\\\=~\frac{-1}{(x+2).(x+2)}\\\\\\=~\boxed{-\frac{1}{(x+2)^2}}$