lim x 0 + tan x tan x - sin x sin x x 3 x Equals

Question

$\lim_{x \to 0^{+}} \frac{\tan x \sqrt{\tan x} - \sin x \sqrt{\sin x}}{x^{3} \sqrt{x}} Equals$

Smart Achievers · Accepted Answer

(1)

$\lim_{x \to 0} \frac{{(\tan x)}^{3 / 2} [1 - {(\cos x)}^{3 / 2}]}{x^{3 / 2} \cdot x^{2}}$

$= 1 \times \lim_{x \to 0} \frac{1 - \cos^{3} x}{x^{2}} \cdot \frac{1}{1 + {(\cos x)}^{3 / 2}}$

$= \frac{1}{2} \cdot \frac{1}{2} (1 + \cos x + \cos^{2} x) = \frac{3}{4}$

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