lim _{x rightarrow 0} frac{tan ^4 x-sin ^4 x} | vedclass

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(C) We have the limit: $\lim _{x ightarrow 0} \frac{	an ^4 x-\sin ^4 x}{x^6}$ Factor out $\sin ^4 x$: $\lim _{x ightarrow 0} \frac{\sin ^4 x(\sec

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(C) We have the limit: $\lim _{x ightarrow 0} \frac{	an ^4 x-\sin ^4 x}{x^6}$ Factor out $\sin ^4 x$: $\lim _{x ightarrow 0} \frac{\sin ^4 x(\sec ^4 x-1)}{x^6}$ Using the identity $\sec ^4 x - 1 = (\sec ^2 x - 1)(\sec ^2 x + 1) = 	an ^2 x (\sec ^2 x + 1)$: $\lim _{x ightarrow 0} \frac{\sin ^4 x \cdot 	an ^2 x (\sec ^2 x + 1)}{x^6}$ Rewrite as: $\lim _{x ightarrow 0} \left( \frac{\sin x}{x} ight)^4 \cdot \left( \frac{	an x}{x} ight)^2 \cdot (\sec ^2 x + 1)$ Since $\lim _{x ightarrow 0} \frac{\sin x}{x} = 1$ and $\lim _{x ightarrow 0} \frac{	an x}{x} = 1$: $= (1)^4 \cdot (1)^2 \cdot (\sec ^2 0 + 1) = 1 \cdot 1 \cdot (1 + 1) = 2$

$\lim _{x \rightarrow 0} \frac{\tan ^4 x-\sin ^4 x}{x^6} = $

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