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Problem 3 - Entrance Test

If f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0, determine if f'(0) exists and find its value if it does.

Correct: B

f'(0) = lim (h→0) (f(h) - f(0)) / h = lim (h→0) (h^2 sin(1/h) - 0) / h = lim (h→0) h sin(1/h). Since -1 ≤ sin(1/h) ≤ 1, -|h| ≤ h sin(1/h) ≤ |h|. By the Squeeze Theorem, lim (h→0) h sin(1/h) = 0. Thus, f'(0) = 0.