Question

Derivadas através metodo de limites helpeeee

f(x) = 3/x
a=3

Answer 1

Calcular a derivada da função

$f(x)=\dfrac{3}{x}$

no ponto $a=3:$

_____________

$f'(a)=\underset{x\to a}{\mathrm{\ell im}}~\dfrac{f(x)-f(a)}{x-a}$

( se o limite existir )


Para $a=3,$ temos

$f'(3)=\underset{x\to 3}{\mathrm{\ell im}}~\dfrac{f(x)-f(3)}{x-3}\\\\\\ =\underset{x\to 3}{\mathrm{\ell im}}~\dfrac{\frac{3}{x}-\frac{3}{3}}{x-3}\\\\\\ =\underset{x\to 3}{\mathrm{\ell im}}~\dfrac{\frac{3}{x}-1}{x-3}\\\\\\ =\underset{x\to 3}{\mathrm{\ell im}}~\dfrac{\frac{3}{x}-1}{x-3}\cdot \dfrac{x}{x}$

$=\underset{x\to 3}{\mathrm{\ell im}}~\dfrac{\left(\frac{3}{x}-1 \right )\cdot x}{(x-3)\cdot x}\\\\\\ =\underset{x\to 3}{\mathrm{\ell im}}~\dfrac{3-x}{(x-3)\cdot x}\\\\\\ =\underset{x\to 3}{\mathrm{\ell im}}~\dfrac{(x-3)\cdot (-1)}{(x-3)\cdot x}\\\\\\ =\underset{x\to 3}{\mathrm{\ell im}}~\dfrac{-1}{x}\\\\\\ =-\dfrac{1}{3}$


Bons estudos! :-)