Evaluate the Limit limit as x approaches 0 of sin(1/x)

Problem

(lim_x→0)(sin(1/x))

Solution

1. Analyze the behavior of the inner function 1/x as x approaches 0 As x gets closer to 0 from the right, 1/x approaches ∞ and as x gets closer to 0 from the left, 1/x approaches −∞

2. Observe the behavior of the sine function. The function sin(θ) oscillates infinitely many times between −1 and 1 as θ approaches ∞ or −∞

3. Determine the limit. Because the value of sin(1/x) continues to oscillate between −1 and 1 and does not settle on a single value as x approaches 0 the limit does not exist.

Final Answer

(lim_x→0)(sin(1/x))=Does Not Exist

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