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If the limit as x approaches 0 of [ax - (e^(4x) - 1)] / [ax(e^(4x) - 1)] exists and equals b, find the value of a - 2b.

1. 5
2. 5
3. 5
4. 5

Correct answer: 5

Solution

As x->0, e^(4x)-1 ~ 4x, so denominator ~ ax*4x = 4ax². For the limit to exist (be finite and non-zero), the numerator must also be O(x²). Numerator = ax-(e^(4x)-1) ~ (a-4)x, which is O(x) unless a=4. Setting a=4: numerator = 4x-(e^(4x)-1) ~ 4x-(4x+8x²+...) = -8x². Denominator ~ 4*4*x²=16x². Limit b = -8/16 = -1/2. Therefore a-2b = 4-2(-1/2) = 4+1 = 5.

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