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Evaluate the limit as x approaches 0: ((1-cos 2x)(3+cos x))/(xtan 4x).

1. -1/4
2. 1/2
3. 1
4. 2

Correct answer: 2

Solution

As x approaches 0, the expression can be simplified using Taylor series expansions for cosine and tangent. The limit evaluates to 2 after applying L'Hôpital's rule or recognizing that the leading terms in the numerator and denominator yield a limit of 2.

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