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

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Multiple Choice

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

Explanation:
The behavior of the expression changes with the sign of x. For positive x, |x|/x = 1, while for negative x, |x|/x = -1. Therefore the limit as x approaches zero from the right is 1, but from the left it is -1. Since these two one-sided limits don’t match, the two-sided limit as x approaches zero does not exist. (The expression isn’t defined at zero, but the limit looks at behavior as you approach from either side.)

The behavior of the expression changes with the sign of x. For positive x, |x|/x = 1, while for negative x, |x|/x = -1. Therefore the limit as x approaches zero from the right is 1, but from the left it is -1. Since these two one-sided limits don’t match, the two-sided limit as x approaches zero does not exist. (The expression isn’t defined at zero, but the limit looks at behavior as you approach from either side.)

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