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

Problem

(lim_x→0)(sin(π/x))

Solution

1. Analyze the behavior of the inner function π/x as x approaches 0 As x gets closer to 0 from the right, π/x approaches ∞ and as x gets closer to 0 from the left, π/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 behavior. Since the value of sin(π/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(π/x))=Does Not Exist

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