Question

lim x --> 0 (cos2x - cos3x)/ x^2

Se alguém puder me ajudar, eu preciso da resolução SEM o uso da Regra de L'Hospital. desde já, eu agradeço!

Answer 1

A ideia vai ser fazer aparecer o limite fundamental :

$\displaystyle \lim_{\text k \to 0}\frac{\text{sen(k)}}{\text k} = 1$

Vamos usar uma das fórmulas de Werner :

$\displaystyle \text{cos(p)} - \text{cos(q) } = - 2\text{sen}(\frac{\text p -\text q}{2}).\text{sen}(\frac{\text p+\text q}{2})$

Temos :

$\displaystyle \lim_{\text x\to 0} \frac{(\text{cos(2x)}-\text{cos(3x)})}{\text x^2} \\\\ \underline{\text{aplicando a formula de werner}}: \\\\\\ \lim_{\text x\to 0} \frac{-2\text{sen}(\frac{2\text x-3\text x}{2}).\text{sen}(\frac{2\text x+3\text x}{2})}{\text x^2 }\\\\\\ \lim_{\text x \to 0} \frac{-2\text{sen}(\frac{-\text x}{2}).\text{sen}(\frac{5\text x}{2})}{\text x^2}$

$\displaystyle \lim_{\text x \to 0 }\frac{\displaystyle -\text{sen}(-\frac{\text x}{2}).\text{sen}(\displaystyle \frac{5\text x}{2})}{\displaystyle \frac{\text x^2}{2}}$

A função seno é ímpar, ou seja :

$\text{sen}(-\text x) = -\text{sen(x)}$

substituindo :

$\displaystyle \lim_{\text x \to 0 }\frac{\displaystyle \text{sen}(\frac{\text x}{2}).\text{sen}(\displaystyle \frac{5\text x}{2})}{\displaystyle \frac{\text x^2}{2}} \\\\ \underline{\text{separando os limites}}: \\\\ \lim_{\text x\to 0}\frac{\text{sen}(\displaystyle \frac{\text x}{2})}{\displaystyle \frac{\text x}{2}} . \lim_{\text x \to 0}\frac{\text{sen}(\displaystyle \frac{5\text x}{2})}{\text x} \\\\\\ 1.\lim_{\text x\to 0}\frac{\displaystyle \text{sen}(\frac{5\text x}{2})}{\text x}$

Vamos multiplicar o numerador e o denominador por 5/2 :

$\displaystyle \lim_{\text x \to 0 }\frac{\displaystyle \frac{5}{2}.\text{sen}(\frac{5\text x}{2})}{\displaystyle \frac{5\text x}{2}} \\\\\\ \frac{5}{2}.\lim_{\text x \to 0 }\frac{\displaystyle \text{sen}(\frac{5\text x}{2})}{\displaystyle \frac{5\text x}{2}} = \frac{5}{2}.1$

Portanto :

$\huge\boxed{\displaystyle \lim_{\text x \to 0} \ \frac{\text{cos(2x)}-\text{cos(3x)}}{\text x^2} = \frac{5}{2}}\checkmark$