Which of the following is a correct notation for the left-hand limit of f as x approaches a?

• A. lim_{x->a^+} f(x) = L.
• B. lim_{x->a} f(x) = L.
• C. lim_{x->a^-} f(x) does not exist.
• D. lim_{x->a^-} f(x) = L.

Answer: (D)

The left-hand limit describes how f(x) behaves as x approaches a from values less than a. This direction is shown with a minus sign in the limit notation: lim_{x->a^-} f(x). If those values approach a single value L, you write lim_{x->a^-} f(x) = L. This exactly captures the idea of approaching from the left and assigning the resulting value. The other forms refer to different ideas: lim_{x->a^+} f(x) uses the right-hand side, lim_{x->a} f(x) is the two-sided limit requiring both sides to agree, and saying a limit does not exist is a statement about existence, not the left-hand notation itself. So lim_{x->a^-} f(x) = L is the correct way to denote the left-hand limit.