Question

Lim sen³x/x² = ?
x->0

Answer 1

O limite fundamental trigonométrico afirma que: $\displaystyle\lim_{x \to 0} {\frac{senx}{x}}$=0

Logo, façamos aparecer um $x^3$ no denominador:
$\displaystyle\lim_{x \to 0} {\frac{sen^3x}{x^2}}$, multiplicando e dividindo por $x$ obtemos:
$\displaystyle\lim_{x \to 0} {\frac{x.sen^3x}{x^3}}$

=$\displaystyle\lim_{x \to 0} {x} \cdot [\displaystyle\lim_{x \to 0} {\cdot{\frac{senx}{x}}]^3$

=0.0^3=0

Answer 2

Olá!

Para resolver esse limite devemos saber que: $\mathsf{\lim_{x \to 0} \frac{\sin x}{x} = 1}$.
 
 Daí, temos que:

$\\ \mathsf{\lim_{x \to 0} \frac{\sin^3 x}{x^2} =} \\\\\\ \mathsf{\lim_{x \to 0} \frac{\sin^3 x}{x^2} \cdot \frac{x}{x}=} \\\\\\ \mathsf{\lim_{x \to 0} \frac{\sin^3 x}{x^3} \cdot x =} \\\\\\ \mathsf{\lim_{x \to 0} \left (\frac{\sin x}{x} \right )^3 \cdot x =} \\\\\\ \mathsf{\lim_{x \to 0} \left (\frac{\sin x}{x} \right )^3 \cdot \lim_{x \to 0} \ x =} \\\\\\ \mathsf{1^3 \cdot 0 =} \\\\ \mathsf{1 \cdot 0 =} \\\\ \boxed{\mathsf{0}}$