Question

Calcule lim f(x + h) − f(x)/h, sendo f dada por b) f(x) = −2x^{3} + 3x
h→0

Answer 1

Resposta:

manda a foto da tarefa pra mim ver direito

Answer 2

$\text{f(x)}=-2\text x^3+3\text x$

Aplicando o limites :

$\displaystyle \lim_{\text h\to0} \frac{\text{f(x+h)}-\text{f(x)}}{\text h} \\\\\\ \lim_{\text h\to 0} \frac{-2(\text{x+h})^3+3(\text{x+h})-[\ -2\text x^3+3\text x\ ] }{\text h} \\\\\\ \lim_{\text h\to0 } =\frac{-2(\text x^3+3\text x^2.\text h+3\text x.\text h^2+\text h^3) + 3\text x+3\text h +2\text x^3-3\text x}{\text h}$

$\displaystyle \lim_{\text h\to0 } =\frac{-2\text x^3-6\text x^2.\text h-6\text x.\text h^2-2\text h^3 + 3\text x+3\text h +2\text x^3-3\text x}{\text h} \\\\\\ \lim_{\text h\to0 } =\frac{-6\text x^2.\text h-6\text x.\text h^2-2\text h^3 +3\text h }{\text h} \\\\\\ \lim_{\text h\to0 } =\frac{\text h(-6\text x^2-6\text x.\text h-2\text h^2 +3) }{\text h} \\\\\\ \lim_{\text h\to0 } =-6\text x^2-6\text x.\text h-2\text h^2 +3$

$\displaystyle \underline{\text{fazendo h = 0}} : \\\\\\ -6.\text x^2-6\text {x.0}-2.\text 0^2+3 \\\\ -6\text x^2 +3$

Portanto :

$\huge\boxed{\displaystyle \lim_{\text h\to0}\frac{\text{f(x+h)}-\text{f(x)}}{\text h} = -6\text x^2+3}\checkmark$

comentário :  

esse limite é a derivada pela definição.