Question

Encontre o limites trigonométricos de sen3x/2x quando x tende a 0

Answer 1

im senx.sen3x.sen5x / tan2x.tan4x.tan6x
x -> 0

Lim (senx/x).(sen3x/3x).(sen5x/5x).x.3x.5x / (tan2x/2x).(tan4x/4x). (tan6x/6x).2x.4x.6x
x -> 0

We know this :
Lim senx/x = 1
x -> 0

Lim sen3x/3x = 1
x -> 0

Lim sen5x/5x = 1
x -> 0

Lim tan2x/2x = 1
x -> 0

Lim tan4x/4x = 1
x -> 0

Lim tan6x/6x = 1
x -> 0

So that :
Lim x.3x.5x / 2x.4x.6x
x -> 0

= 1.3.5 / 2.4.6
= 5/16

Answer 2

Olá,

$lim_{x - > 0}( \frac{ \sin(3x) }{2x} )$

Simplifique usando:
$lim_{x - > a}(c \times f(x)) = c \times lim_{x - > af(x)}$

Daí, obtemos:

$\frac{1}{2} lim_{x - > 0}( \frac{ \sin(3x) }{x} )$

Aplique a regra de L'Hopital.

$\frac{1}{2} lim_{x - > 0}( \frac{ \cos(3x) \times 3 }{1} )$

Simplifique.

$\frac{1}{2} lim_{x - > 0}(3 \cos(3x) )$

Insira o valor.

$\frac{1}{2} \times 3 \cos(3 \times 0)$

Simplifique.

$\frac{3}{2}$

Espero ter te ajudado!