Compute lim x->0 |x|.

Compute lim x->0 |x|.

• A. 0
• B. 1
• C. -1
• D. Infinity

Answer: (A)

At small x, |x| represents how far x is from zero. As x approaches zero from either side, that distance shrinks toward zero. If x is nonnegative, |x| = x; if x is negative, |x| = -x. In both cases, as x → 0, the value |x| → 0. In epsilon-delta terms, for any epsilon > 0, choosing delta = epsilon ensures that whenever 0 < |x| < delta we have |x| < epsilon. Therefore the limit equals 0. The other values, like 1, -1, or infinity, do not match the behavior of |x| near zero because |x| is always nonnegative and tends to zero, not to a positive constant or to infinity.

At small x, |x| represents how far x is from zero. As x approaches zero from either side, that distance shrinks toward zero. If x is nonnegative, |x| = x; if x is negative, |x| = -x. In both cases, as x → 0, the value |x| → 0. In epsilon-delta terms, for any epsilon > 0, choosing delta = epsilon ensures that whenever 0 < |x| < delta we have |x| < epsilon. Therefore the limit equals 0. The other values, like 1, -1, or infinity, do not match the behavior of |x| near zero because |x| is always nonnegative and tends to zero, not to a positive constant or to infinity.