limθ→0⁡3tan⁡θ−tan⁡3θ2θ3=

limθ→0⁡3tan⁡θ−tan⁡3θ2θ3=limθ→0⁡3tan⁡θ−3tan⁡θ−tan3⁡θ1−3tan2⁡θ2θ3=limθ→0⁡−8tan3⁡θ2θ31−3tan2⁡θ=−82(1−0)=−4

limθ→0⁡3tan⁡θ−tan⁡3θ2θ3=

limθ→0⁡3tan⁡θ−tan⁡3θ2θ3=limθ→0⁡3tan⁡θ−3tan⁡θ−tan3⁡θ1−3tan2⁡θ2θ3=limθ→0⁡−8tan3⁡θ2θ31−3tan2⁡θ=−82(1−0)=−4

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$\lim_{θ \to 0} \frac{3 \tan θ - \tan 3 θ}{2 θ^{3}} =$

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1/4

-4

3/4

answer is D.

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Detailed Solution

$\lim_{θ \to 0} \frac{3 \tan θ - \tan 3 θ}{2 θ^{3}} = \lim_{θ \to 0} \frac{3 \tan θ - \frac{3 \tan θ - \tan^{3} θ}{1 - 3 \tan^{2} θ}}{2 θ^{3}} = \lim_{θ \to 0} \frac{- 8 \tan^{3} θ}{2 θ^{3} (1 - 3 \tan^{2} θ)} = - \frac{8}{2 (1 - 0)} = - 4$