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Evaluate the limit as x approaches 0: ((1+tan x)/(1+sin x))^(csc x).

1. 1/e
2. 1
3. e
4. e²

Correct answer: 1

Solution

ln of base = ln(1+tanx)-ln(1+sinx) ~ (tanx - sinx) ~ x^3/2. Multiplying by cosec x ~ 1/x gives ~ x^2/2 -> 0, so the limit is e^0 = 1.

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