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

Problem

(lim_x→0)(cos(1/x))

Solution

1. Analyze the behavior of the inner function 1/x as x approaches 0 As x gets closer to 0 the value of 1/x increases without bound toward ∞ (from the right) or decreases toward −∞ (from the left).

2. Observe the behavior of the cosine function for large inputs. The function cos(u) oscillates infinitely many times between −1 and 1 as u approaches infinity or negative infinity.

3. Determine the limit by considering the oscillation. Because the values of cos(1/x) continue to fluctuate between −1 and 1 and do not settle on a single finite value as x approaches 0 the limit does not exist.

Final Answer

(lim_x→0)(cos(1/x))=Does Not Exist

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