Uniqueness theorem for limits

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Statement

Two-sided limit

Suppose f is a function and c is a point such that f is defined on both the immediate left and the immediate right of c. The uniqueness theorem for limits states that if the limit of f exists at c (in the sense of existence as a finite real number) then it is unique. In other words:

If \lim_{x \to c} f(x) = L and \lim_{x \to c} f(x) = M, then L = M.

Left hand limit

Suppose f is a function and c is a point such that f is defined on the immediate left of c. The uniqueness theorem for left hand limits states that if the left hand limit of f exists at c (in the sense of existence as a finite real number) then it is unique. In other words:

If \lim_{x \to c^-} f(x) = L and \lim_{x \to c^-} f(x) = M, then L = M.

Right hand limit

Suppose f is a function and c is a point such that f is defined on the immediate right of c. The uniqueness theorem for right hand limits states that if the right hand limit of f exists at c (in the sense of existence as a finite real number) then it is unique. In other words:

If \lim_{x \to c^+} f(x) = L and \lim_{x \to c^+} f(x) = M, then L = M.