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Let f(x) = (integral from x to x² of cos(t) dt + x) / (2x). Evaluate the correct limit(s) among the following.

1. lim x->0 f(x) = 0
2. lim x->0 f(x) = 1
3. lim x->1 f(x) = 1/2
4. lim x->1 f(x) = 1

Correct answer: lim x->0 f(x) = 1

Solution

f(x) = [integral(x to x², cos t dt) + x] / (2x). At x->0: integral(x to x², cos t dt) ~ cos(0)(x² - x) = x² - x. So numerator ~ x² - x + x = x² and denominator = 2x. Limit = x²/(2x) = x/2 -> 0. Wait, let me recompute: numerator = (x² - x)cos(c) + x for some c. More carefully, integral = sin(x²) - sin(x). As x->0: sin(x²) ~ x², sin(x) ~ x. Numerator = x² - x + x = x². Limit = x²/(2x) = x/2 -> 0. At x->1: sin(1) - sin(1) + 1 = 1. Denominator = 2. Limit = 1/2.

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