$lim_{x \to 0} \frac{(\tan x - \sin x) (\tan x + \sin x)}{x^{4}}$

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Answer

Apply L'Hopital's rule to find the limit as x approaches 0: $lim_{x \to 0} \frac{a n^{2} x - s i n^{2} x}{x^{4}} = 1$

Steps

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

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

limx→0(tan⁡x−sin⁡x)(tan⁡x+sin⁡x)x4

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$lim_{x \to 0} \frac{(\tan x - \sin x) (\tan x + \sin x)}{x^{4}}$