Question

Calcule o seguinte limite trigonométrico:

lim (x - sen3x)/(x+sen2x)
x - 0

Answer 1

Utilizando as propriedades do limite, podemos transformar a expressão na forma do limite fundamental trigonométrico.

$\displaystyle \mathsf{ \lim_{x \to 0} \frac{x-sen3x}{x+sen2x}}\displaystyle \mathsf{= \lim_{x \to 0} \frac{x \div x}{x+sen2x \div x}-\lim_{x \to 0}\frac{sen3x \div x }{x+sen2x \div x}}}\\\\ \mathsf{= \lim_{x \to 0} \frac{1}{1+\frac{sen2x}{x} }-\lim_{x \to 0}\frac{\frac{sen3x}{x}}{1+\frac{sen2x}{x} }}}\\\\ \mathsf{= \lim_{x \to 0} \frac{1}{1+\frac{sen2x}{x} }-\frac{\lim_{x \to 0}\frac{sen3x}{x}}{\lim_{x \to 0}1+\frac{sen2x}{x} }}}$

$\mathsf{=\frac{1}{1+2.1}-\frac{3.1}{1 + 2.1}} \mathsf{=\frac{1}{1+2}-\frac{3}{1 + 2}} \mathsf{=\frac{1}{3}-\frac{3}{3}}\mathsf{=-\frac{2}{3}}$

Resposta: -2/3