Question

qual a resposta da questão ??

Answer 1

Explicação passo-a-passo:

$\sf lim_{x~\to~0}~\Big(\dfrac{tg~5x}{x}\Big)$

$\sf =lim_{x~\to~0}~\Big(\dfrac{\frac{sen~5x}{cos~5x}}{x}\Big)$

$\sf =lim_{x~\to~0}~\Big(\dfrac{sen~5x}{cos~5x}\cdot\dfrac{1}{x}\Big)$

$\sf =lim_{x~\to~0}~\Big(\dfrac{sen~5x}{x}\cdot\dfrac{1}{cos~5x}\Big)$

$\sf =lim_{x~\to~0}~\Big(\dfrac{sen~5x}{x}\Big)\cdot\lim_{x~\to~0}~\Big(\dfrac{1}{cos~5x}\Big)$

$\sf =lim_{x~\to~0}~\Big(\dfrac{5\cdot sen~5x}{5x}\Big)\cdot\lim_{x~\to~0}~\Big(\dfrac{1}{cos~5x}\Big)$

$\sf =5\cdot lim_{x~\to~0}~\Big(\dfrac{sen~5x}{5x}\Big)\cdot\lim_{x~\to~0}~\Big(\dfrac{1}{cos~5x}\Big)$

=> Limite fundamental:

$\sf lim_{x~\to~0}~\Big(\dfrac{sen~x}{x}\Big)=1$

$\sf \Rightarrow~lim_{x~\to~0}~\Big(\dfrac{sen~5x}{5x}\Big)=1$

Logo:

$\sf lim_{x~\to~0}~\Big(\dfrac{tg~5x}{x}\Big)$

$\sf =5\cdot lim_{x~\to~0}~\Big(\dfrac{sen~5x}{5x}\Big)\cdot\lim_{x~\to~0}~\Big(\dfrac{1}{cos~5x}\Big)$

$\sf =5\cdot1\cdot\lim_{x~\to~0}~\Big(\dfrac{1}{cos~5\cdot0}\Big)$

$\sf =5\cdot\lim_{x~\to~0}~\Big(\dfrac{1}{cos~0}\Big)$

$\sf =5\cdot\lim_{x~\to~0}~\Big(\dfrac{1}{1}\Big)$

$\sf =5\cdot\lim_{x~\to~0}~1$

$\sf =5\cdot1$

$\sf =\red{5}$