Question

Calculo 1 limite


$\lim_{h \to \ 0 } \frac{(x + h)^3 -x^3 }{h}$

Answer 1

Olá



$\displaystyle\mathsf{ \lim_{h \to 0}~ \frac{(x+h)^3-x^3}{h} ~=~ \frac{0}{0} }$



Desenvolve o termo (x+h)³


(x+h)³ = (x+h)(x+h)²

(x+h)³ = (x+h)(x² + 2xh + h²)

(x+h)³ = x³ + 3x²h + 3xh² + h³



Substitui no limite



$\displaystyle\mathsf{ \lim_{h \to 0}~ \frac{x^3+3x^2h+3xh^2+h^3-x^3}{h} }$



Simplifica


$\displaystyle\mathsf{ \lim_{h \to 0}~ \frac{\diagup\!\!\!\!x^3+3x^2h+3xh^2+h^3-\diagup\!\!\!\!x^3}{h} }\\\\\\\\\mathsf{ \lim_{h \to 0}~ \frac{3x^2h+3xh^2+h^3}{h} }$



Coloca o 'h' em evidencia 


$\displaystyle\mathsf{ \lim_{h \to 0}~ \frac{h(3x^2+3xh+h^2)}{h} }$



Simplifica


$\displaystyle\mathsf{ \lim_{h \to 0}~ \frac{\diagup\!\!\!\!h(3x^2+3xh+h^2)}{\diagup\!\!\!\!h} }\\\\\\\\\displaystyle\mathsf{ \lim_{h \to 0}~ {3x^2+3xh+h^2~=~3x^2+3x(0)+0^2}~=~\boxed{\mathsf{3x^2} }}\\\\\\\\\\\boxed{\displaystyle\mathsf{ \lim_{h \to 0}~ \frac{(x+h)^3-x^3}{h}~=~3x^2 }}$