Trigonometria
Aula 10 e exercícios - Limite notável
Aula com exemplos
Exercícios resolvidos
10.1)
Calcule, se existir, cada um dos seguintes limites:
 (a) $\displaystyle\mathop
{\lim }\limits_{x \to 0} \frac{{{\rm sen } (3x)}}{x}$; |
(b) $\displaystyle\mathop
{\lim }\limits_{x \to 0} \frac{4x}{{\rm sen }(6x)}$; |
(c) $\displaystyle\mathop
{\lim }\limits_{x \to 0} \frac{{{\rm tg } (3x)}}{x}$; |
| (d) $\displaystyle\mathop
{\lim }\limits_{x \to \pi } \frac{{{\rm sen } x}}{{\pi - x}}$; |
(e) $\displaystyle\mathop
{\lim }\limits_{x \to + \infty } \left(\frac{x}{2}{\rm sen } \frac{\pi }{x}\right)$; |
(f)
$\displaystyle\mathop {\lim }\limits_{x \to 0} \frac{{{\rm sen } x}}{{{e^x} - 1}}$; |
| (g) $\displaystyle\mathop
{\lim }\limits_{x \to {{\frac{\pi }{3}}^ + }} \frac{{{{\cos }^2}(2x)}}{{\cos x - \frac{1}{2}}}$ ; |
(h) $\displaystyle\mathop
{\lim }\limits_{x \to 0} \frac{{{\rm sen } (3x)}}{{{e^{5x}} - 1}}$; |
(i) $\displaystyle\mathop
{\lim }\limits_{x \to {\pi ^ + }} \frac{3}{{{\rm sen } x}}$. |
1.2)
Determine o valor real de $k$ de modo que a função definida por \[f\left( x \right) = \left\{ \begin{array}{ll}e^{2x}+k &{\rm{ se }} \;x \leq 0 \\\displaystyle\frac{{\rm sen }(2x)}{x} &{\rm{se }}\; x >0 \\\end{array}
\right.\] seja contínua em $x=0$.