\begin{aligned} \lim_{x \to 0} \ \frac{\sqrt{1+\tan x} - \sqrt{1+\sin x}}{x^3} &= \lim_{x \to 0} \ \frac{\sqrt{1+\tan x} - \sqrt{1+\sin x}}{x^3} \cdot \frac{\sqrt{1+\tan x} + \sqrt{1+\sin x}}{\sqrt{1+\tan x} + \sqrt{1+\sin x}} \\[8pt] &= \lim_{x \to 0} \ \frac{(1+\tan x) - (1+\sin x)}{x^3 \ (\sqrt{1+\tan x} + \sqrt{1+\sin x})} \\[8pt] &= \lim_{x \to 0} \ \frac{\tan x - \sin x}{x^3 \ (\sqrt{1+\tan x} + \sqrt{1+\sin x})} \\[8pt] &= \lim_{x \to 0} \ \frac{\tan x - \sin x}{x^3} \cdot \lim_{x \to 0} \ \frac{1}{(\sqrt{1+\tan x} + \sqrt{1+\sin x})} \\[8pt] &= \lim_{x \to 0} \ \frac{\frac{\sin x}{\cos x} - \sin x}{x^3} \cdot \frac{1}{(\sqrt{1+\tan 0} + \sqrt{1+\sin 0})} \\[8pt] &= \lim_{x \to 0} \ \frac{\sin x-\sin x \cos x}{x^3 \cos x} \cdot \frac{1}{\sqrt{1} + \sqrt{1}} \\[8pt] &= \lim_{x \to 0} \ \frac{\sin x (1-\cos x)}{x^3 \cos x} \cdot \frac{1}{2} \\[8pt] &= \frac{1}{2} \cdot \lim_{x \to 0} \ \frac{\sin x \ (2 \sin^2 \frac{1}{2}x)}{x^3 \cos x} \\[8pt] &= \frac{1}{2} \cdot \lim_{x \to 0} \ \frac{2\sin x}{x} \cdot \lim_{x \to 0} \ \frac{\sin^2 \frac{1}{2}x}{x^2} \cdot \lim_{x \to 0} \ \frac{1}{\cos x} \\[8pt] &= \frac{1}{2} \cdot 2 \cdot \left(\frac{1}{2}\right)^2 \cdot \frac{1}{\cos 0} = \frac{1}{4} \end{aligned}