Question

Calcule o seguinte limite trigonométrico

Answer 1

$\displaystyle \lim_{\text x \to 0 }\ \frac{3.\text x^2 }{\text{tg(x).sen(x)}} \\\\\\ 3.\lim_{\text x \to 0 } \ \frac{\text x}{\text{sen(x)}}\ .\ \lim_{\text x \to 0 } \ \frac{x}{\text{tg(x)}} \\\\\\ 3.\lim_{\text x \to 0 } \ (\frac{\text{sen(x)}}{\text x})^{-1} \ . \ \lim_{\text x \to 0 } \ \frac{\text x}{\frac{\displaystyle \text{sen(x)}}{\text{cos(x)}}}$

continuando a desenvolver :

$\displaystyle 3.[\ \lim_{\text x \to 0 } \ \frac{\text{sen(x)}}{\text x} \ ]^{-1} \ . \ \lim_{\text x \to 0 } \ \frac{\displaystyle \text x.\text{cos(x)}}{\text{sen(x)}}$

$\displaystyle 3.[\ \lim_{\text x \to 0 } \ \frac{\text{sen(x)}}{\text x} \ ]^{-1} \ . \lim_{\text x \to 0 } \ \frac{\text{x}}{\text {sen(x)}} . \ \lim_{\text x \to 0 } \text{cos(x)} \\\\\\ 3. [\ \lim_{\text x \to 0 } \ \frac{\text{sen(x)}}{\text x} \ ]^{-1} \ .[\ \lim_{\text x \to 0 } \ \frac{\text{sen(x)}}{\text x} \ ]^{-1} \ .\ \lim_{\text x \to 0 } \text{cos(x)} \\\\\\ 3.1^{-1} . 1^{-1}. \text{cos(0)} = 3$

Portanto :

$\huge\boxed{\displaystyle \lim_{\text x \to 0 }\ \frac{3.\text x^2 }{\text{tg(x).sen(x)}} = 3 }\checkmark$