lim _{x rightarrow 0} frac{4 sin x sin ^{2} frac{x}{2}}{x^{3}}

Question

Accepted Answer

Step:1 ：Find the derivatives of the numerator and the denominator: (frac{d}{dx}(4 sin x sin^2(frac{x}{2})) = 4sin(frac{x}{2})^2cos(x) + 4sin(frac{x}{2})sin(x)cos(frac{x}{2})) and (frac{d}{dx}(x^3) = 3x^2)Step:2 ：Apply L&#x27;Hopital&#x27;s rule and find the limit as x approaches 0: (lim_{x to 0} frac{4sin(frac{x}{2})^2cos(x) + 4sin(frac{x}{2})sin(x)cos(frac{x}{2})}{3x^2} = boxed{1})

Answer

Step: 1 ：Find the derivatives of the numerator and the denominator: (frac{d}{dx}(4 sin x sin^2(frac{x}{2})) = 4sin(frac{x}{2})^2cos(x) + 4sin(frac{x}{2})sin(x)cos(frac{x}{2})) and (frac{d}{dx}(x^3) = 3x^2)Step: 2 ：Apply L&#x27;Hopital&#x27;s rule and find the limit as x approaches 0: (lim_{x to 0} frac{4sin(frac{x}{2})^2cos(x) + 4sin(frac{x}{2})sin(x)cos(frac{x}{2})}{3x^2} = boxed{1})

$\lim _{x \rightarrow 0} \frac{4 \sin x \sin ^{2} \frac{x}{2}}{x^{3}}$

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