A piecewise function is defined by for and for

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

A piecewise function gg is defined by g(x)=2g(x)=2 for x<3x<3 and g(x)=−1g(x)=-1 for x≥3x\ge 3. Which statement best accounts for why lim⁡x→3g(x)\displaystyle \lim_{x\to 3}g(x) does not exist? (a) The limit fails because g(3)≠2g(3)\neq 2, so the function value and the approaching value are different. (b) The limit fails simply because gg is not continuous at x=3x=3. (c) The limit fails because the left‐hand and right‐hand limits approach different finite numbers. (d) The limit fails because the function becomes unbounded as xx approaches 33.

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