What is the limit of |x| as x -> 0?

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Multiple Choice

What is the limit of |x| as x -> 0?

Explanation:
Think of what |x| represents near zero: the distance from x to zero. As x gets closer to zero, that distance gets smaller and approaches zero. So the limit of |x| as x approaches 0 is 0. You can see this rigorously with the epsilon-delta idea: for any ε > 0, if |x| < ε (which means x is within ε of 0), then ||x| − 0| = |x| < ε. Hence the limit is 0. The graph is a V with its tip at the origin, showing the distance to zero goes to zero from both sides. This isn’t 1 or infinity, and the limit exists and equals 0.

Think of what |x| represents near zero: the distance from x to zero. As x gets closer to zero, that distance gets smaller and approaches zero. So the limit of |x| as x approaches 0 is 0.

You can see this rigorously with the epsilon-delta idea: for any ε > 0, if |x| < ε (which means x is within ε of 0), then ||x| − 0| = |x| < ε. Hence the limit is 0. The graph is a V with its tip at the origin, showing the distance to zero goes to zero from both sides. This isn’t 1 or infinity, and the limit exists and equals 0.

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