For lim_{x->a} f(x) to exist as a finite limit, what must be true of the one-sided limits?

For lim_{x->a} f(x) to exist as a finite limit, what must be true of the one-sided limits?

• A. lim_{x->a^+} f(x) and lim_{x->a^-} f(x) exist and are equal to the same finite L.
• B. Either one of the one-sided limits exists; no need to be equal.
• C. The function must be continuous at a.
• D. The limit must be infinite.

Answer: (A)

The main idea is that a finite two-sided limit as x approaches a can only exist if approaching from the left and from the right both settle to the same finite value. In other words, the left-hand limit and the right-hand limit must both exist and be equal to the same number L. If either side fails to exist, or if the two sides converge to different values, there is no finite limit at a. Note that the limit concerns the behavior of f(x) as x gets arbitrarily close to a, not the actual value of f at a. So the function can be discontinuous at a (or even undefined there) and the limit can still exist and equal L. Also, if the limit would be infinite, that is not a finite limit, so that scenario doesn’t satisfy the requirement. For intuition, if both sides approach 3 as x → a, the limit exists and equals 3. If the left side approaches 2 and the right side 4, the two-sided limit does not exist. If both sides approach 0 but f(a) is something else, the limit is still 0.

The main idea is that a finite two-sided limit as x approaches a can only exist if approaching from the left and from the right both settle to the same finite value. In other words, the left-hand limit and the right-hand limit must both exist and be equal to the same number L. If either side fails to exist, or if the two sides converge to different values, there is no finite limit at a.

Note that the limit concerns the behavior of f(x) as x gets arbitrarily close to a, not the actual value of f at a. So the function can be discontinuous at a (or even undefined there) and the limit can still exist and equal L. Also, if the limit would be infinite, that is not a finite limit, so that scenario doesn’t satisfy the requirement.

For intuition, if both sides approach 3 as x → a, the limit exists and equals 3. If the left side approaches 2 and the right side 4, the two-sided limit does not exist. If both sides approach 0 but f(a) is something else, the limit is still 0.