Determine lim_{x→0} |sin(1/x)|.

Determine lim_{x→0} |sin(1/x)|.

• A. 0
• B. π/2
• C. DNE
• D. 1

Answer: (C)

As x approaches zero, the expression inside the sine, 1/x, grows without bound, so sin(1/x) keeps oscillating between -1 and 1 without settling to a single value. Taking the absolute value confines the range to [0, 1], but the limit would have to be the same number no matter how x approaches zero. You can choose sequences that push the value to 0 (for instance, when 1/x = nπ, so sin(1/x) = 0) and sequences that push the value to 1 (for example, when 1/x = π/2 + 2πn, so sin(1/x) = 1). Because different approaches yield different limiting values, the limit does not exist.

As x approaches zero, the expression inside the sine, 1/x, grows without bound, so sin(1/x) keeps oscillating between -1 and 1 without settling to a single value. Taking the absolute value confines the range to [0, 1], but the limit would have to be the same number no matter how x approaches zero. You can choose sequences that push the value to 0 (for instance, when 1/x = nπ, so sin(1/x) = 0) and sequences that push the value to 1 (for example, when 1/x = π/2 + 2πn, so sin(1/x) = 1). Because different approaches yield different limiting values, the limit does not exist.