Evaluate the limit as x approaches 0 of |x|/x.

Evaluate the limit as x approaches 0 of |x|/x.

• A. DNE
• B. 0
• C. 1
• D. -1

Answer: (A)

As x gets close to zero, the expression |x|/x behaves differently on each side: for positive x it equals 1, and for negative x it equals -1. So the limit from the right is 1, while the limit from the left is -1. Because these one-sided limits do not agree, the two-sided limit cannot exist. The function itself isn’t defined at zero, but that doesn’t force a limit; the mismatch of the side limits is what prevents a finite value. Therefore the limit does not exist.

As x gets close to zero, the expression |x|/x behaves differently on each side: for positive x it equals 1, and for negative x it equals -1. So the limit from the right is 1, while the limit from the left is -1. Because these one-sided limits do not agree, the two-sided limit cannot exist. The function itself isn’t defined at zero, but that doesn’t force a limit; the mismatch of the side limits is what prevents a finite value. Therefore the limit does not exist.