Question

Вычислить предел

1. $\lim_{x \to \ 0} \frac{2sin5x}{8x}$
2. $\lim_{x \to \ 0 } \frac{3x-sin2x}{sinx}$

Answer 1

$\lim_{x \to 0} \frac{2 \sin(5x) }{8x} = \lim_{x \to 0}( \frac{ \sin(5x) }{4x} )' = \\ = \lim_{x \to 0} \frac{5 \cos(5x) }{4} = \frac{5 \cos(0) }{4} = \frac{5}{4}$

$\lim_{x \to 0} \frac{3x - \sin(2x) }{ \sin(x) } = \lim_{x \to 0}( \frac{3x - \sin(2x) }{ \sin(x) } )' = \\ = \lim_{x \to 0} \frac{3 - 2 \cos(2x) }{ \cos(x) } = \frac{3 - 2}{1} = 1$