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Answer :
Final answer:
To compute the limit lim x→∞ (1-cos(x)6)/x12 using L'Hôpital's Rule, we can apply the rule twice and simplify the expression to cos(x)10/x10. As x approaches infinity, the cosine term approaches a value between -1 and 1, and x10 increases without bound, leading to a final limit of 0.
Explanation:
To compute the limit limx→∞ (1-cos(x)6)/x12 using L'Hôpital's Rule, we can first rewrite the expression as [(1-cos(x)6)/(x6)2]/(x6)2. Now, we can apply L'Hôpital's Rule twice. Taking the derivative of the numerator and denominator twice, we get (6cos(x)5)2/(6x5)2 which simplifies to cos(x)10/x10. Finally, as x approaches infinity, cos(x) approaches a value between -1 and 1, and x10 increases without bound. Therefore, the limit is 0.
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