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

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: \(\lim_{x \rightarrow 0} \frac{ an^2 x - sin^2 x}{x^4} = \boxed{1}\)