$\lim _{x \rightarrow 0} \frac{(\tan x-\sin x)(\tan x+\sin x)}{x^{4}}$

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Apply L'Hopital's rule to find the limit as x approaches 0: $\lim_{x \rightarrow 0} \frac{ an^2 x - sin^2 x}{x^4} = \boxed{1}$

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Step 1 ：Use the identity $(a - b)(a + b) = a^2 - b^2$ to simplify the expression: $\frac{( an x - sin x)( an x + sin x)}{x^4} = \frac{ an^2 x - sin^2 x}{x^4}$

Step 2 ：Apply L'Hopital's rule to find the limit as x approaches 0: $\lim_{x \rightarrow 0} \frac{ an^2 x - sin^2 x}{x^4} = \boxed{1}$

$\lim _{x \rightarrow 0} \frac{(\tan x-\sin x)(\tan x+\sin x)}{x^{4}}$

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