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exraven
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Homework Statement
Prove:
lim (n→0) {(sin n cos n) / (ⁿ√(1 - tan² 2n + tan⁴ 2n - tan⁶ 2n + ...))} = 1
Homework Equations
The Attempt at a Solution
The denominator is an infinite geometric series, using the sum formula of an infinite geometric series, I simplify the limit:
lim (n→0) {(sin n cos n) / (ⁿ√(1 - tan² 2n + tan⁴ 2n - tan⁶ 2n + ...))}
= lim (n→0) {(sin n cos n) / (ⁿ√(1 / (1 + tan² 2n)))}
After a few tries, i can't figure out what to do with the ⁿ√, rationalizing the denominator won't work because the ⁿ√ will stuck in the numerator. L'hopital is not allowed because this is a high school limit problem. Could anyone give me some hints to solve this ? Any help would be much appreciated.