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Let a be a positive constant. Consider the curves C1: y = e^x and C2: y = e^(a-x). Let S(a) be the area of the region bounded by C1, C2 and the y-axis. Which statement is correct?

1. Lim (a->0) S = 1
2. Lim (a->0) S/a² = 1/4
3. Range of S is [0, infinity)
4. S(a) is neither odd nor even

Correct answer: Lim (a->0) S/a² = 1/4

Solution

S(a) = e^(a/2)... compute area = 2(e^(a/2) - 1) - a...; for small a, S ~ a²/4, so Lim S/a² = 1/4.

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