Is the function f(x) = x^2 sin(1/x) differentiable at x = 0?
Correct Answer: Yes, f(x) is differentiable at x = 0.
- Step 1: Understand the function f(x) = x^2 sin(1/x). This function is defined for all x except x = 0, but we need to check its behavior at x = 0.
- Step 2: To check if f(x) is differentiable at x = 0, we use the limit definition of the derivative: f'(0) = lim (h -> 0) [(f(h) - f(0)) / h].
- Step 3: Calculate f(0). Since f(x) is not defined at x = 0, we define f(0) = 0 for the purpose of this limit.
- Step 4: Substitute f(h) into the limit: f(h) = h^2 sin(1/h). So, we have f'(0) = lim (h -> 0) [(h^2 sin(1/h) - 0) / h].
- Step 5: Simplify the expression: f'(0) = lim (h -> 0) [h sin(1/h)].
- Step 6: Analyze the limit. As h approaches 0, sin(1/h) oscillates between -1 and 1, so h sin(1/h) will approach 0 because h approaches 0.
- Step 7: Conclude that f'(0) = 0. Since the limit exists and is finite, f(x) is differentiable at x = 0.
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