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If $\displaystyle{\lim_{x\to 1^-} f(x) = 5}$ , then $\displaystyle{\lim_{x\to 1} f(x) = 5}$ .

If $\displaystyle{\lim_{x\to 1^-} f(x) = 5}$ , then $\displaystyle{\lim_{x\to 1^+} f(x) = 5}$ .

If $\displaystyle{\lim_{x\to 1} f(x) = 5}$ , then $\displaystyle{\lim_{x\to 1^-} f(x) = 5}$ .

If $\displaystyle{\lim_{x\to 1} f(x) = 5}$ , then $\displaystyle{\lim_{x\to 1^+} f(x) = 5}$ .

Select all true statements. Assume that all the limits are all taken at the same point.

A. If the right-hand limit exists, then the two-sided limit exists.
B. If the left-hand limit exists, then the two-sided limit exists.
C. If the left- and right-hand limits both exist and are equal, then the two-sided limit exists.
D. If the two-sided limit exists, then the left- and right-hand limits both exist and are equal.