Mathematics

Mathematics

Problem

State the relationship that must hold between the one-sided limits lim⁡x→c−f(x)\displaystyle \lim_{x\to c^-}f(x) and lim⁡x→c+f(x)\displaystyle \lim_{x\to c^+}f(x) in order for the two-sided limit lim⁡x→cf(x)\displaystyle \lim_{x\to c}f(x) to exist.
Intuitive explanation

A two-sided limit exists only when the function approaches the same value from both directions. Think of it like a function "converging" to a point: if the left-hand approach and right-hand approach don't meet at the same height, the overall limit doesn't exist—there's a discontinuity or jump. The key insight is that both one-sided limits must exist and be equal for the two-sided limit to be defined.

Intuitive explanation

A two-sided limit exists only when the function approaches the same value from both directions. Think of it like a function "converging" to a point: if the left-hand approach and right-hand approach don't meet at the same height, the overall limit doesn't exist—there's a discontinuity or jump. The key insight is that both one-sided limits must exist and be equal for the two-sided limit to be defined.

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