atan(x) * sin(1/x) goes to zero as x goes to zero. This is because the sine is bounded by 1 in absolute value, and atan(0)=0. Therefore, the answer is ln(2) by continuity of the square root function.
The fact that sin(1/x) does not possess a limit as x approaches 0 is immaterial to this question; it is not necessary to apply the L’Hopital rule.

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