Question

determine o limite da função 3x-5.sen(4x) / 5x+2.sen(8x) quando x tende a zero

Answer 1

Olá. Vamos apenas fazer algumas alterações algébricas, acompanhe:

$\displaystyle \lim_{x \to 0} \, \frac{3x-5 \sin 4x}{5x+ 2 \sin 8x} \\ \\ \\ \lim_{x \to 0} \, \frac{x \cdot (3-\displaystyle \frac{5 \sin 4x}{x})}{x \cdot (5+\displaystyle \frac{2 \sin 8x}{x} )} \\ \\ \\ \lim_{x \to 0} \, \frac{3-\displaystyle \frac{5 \sin 4x}{x}}{5+\displaystyle \frac{2 \sin 8x}{x}} \\ \\ \\ \lim_{x \to 0} \, \frac{3-\displaystyle 5 \cdot \frac{4\sin 4x}{4x}}{5+\displaystyle 2 \cdot \frac{8 \sin 8x}{8x}}$

$\displaystyle \lim_{x \to 0} \, \frac{3-\displaystyle 5 \cdot 4 \cdot \frac{\sin 4x}{4x}}{5+\displaystyle 2 \cdot 8 \cdot \frac{ \sin 8x}{8x}} \\ \\ \\ \lim_{x \to 0} \, \frac{3-5 \cdot 4}{5+ 2 \cdot 8 } = \frac{3-20}{5+16} = \boxed{\boxed{-\frac{17}{21} }}$