Question

2) Utilizando a regra de L'Hospital, calcule os limites
a) limx=0
senx
5x​

Answer 1

Resposta:

$\frac{1}{5}$

Explicação:

$\lim_{x \to \00} \frac{senx}{5x} = \lim_{x \to \00} \frac{cosx}{5} =\frac{cos0}{5} =\frac{1}{5}$

Answer 2

$\Large\boxed{\begin{array}{l}\underline{\rm Sem~a~regra~de~L'Hospital:}\\\displaystyle\sf \lim_{x \to 0}\dfrac{sen(x)}{5x}\\\displaystyle\sf \dfrac{1}{5}\cdot\lim_{x \to 0}\dfrac{sen(x)}{x}\\\sf=\dfrac{1}{5}\cdot1=\dfrac{1}{5}\\\underline{\rm Com~a~regra~de~L'Hospital:}\\\displaystyle\sf \lim_{ x \to 0}\dfrac{sen(x)}{5x}\\\displaystyle\sf\lim_{x \to 0}\dfrac{\frac{d}{dx}~sen(x)}{\frac{d}{dx}~5x}\\\displaystyle\sf \lim_{x \to 0}\dfrac{ cos(x)}{5}=\dfrac{1}{5}\end{array}}$