Problem

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

Solution

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

From Solvely APP
Source: https://solvelyapp.com/problems/21626/

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