If both one-sided limits exist and are not equal, what can be said about the two-sided limit?

If both one-sided limits exist and are not equal, what can be said about the two-sided limit?

• A. The two-sided limit exists.
• B. The two-sided limit equals the left-hand limit.
• C. The two-sided limit equals the right-hand limit.
• D. The two-sided limit does not exist.

Answer: (D)

A two-sided limit requires that the values approached from the left and from the right converge to the same number. If both one-sided limits exist but are not equal, there is no single value that the function approaches from both directions. So, the two-sided limit does not exist in this situation. For example, if as x approaches a from the left the function tends to L and from the right it tends to R with L ≠ R, you can’t assign a single limit value for x approaching a from both sides. That’s why the two-sided limit fails to exist.

A two-sided limit requires that the values approached from the left and from the right converge to the same number. If both one-sided limits exist but are not equal, there is no single value that the function approaches from both directions. So, the two-sided limit does not exist in this situation.

For example, if as x approaches a from the left the function tends to L and from the right it tends to R with L ≠ R, you can’t assign a single limit value for x approaching a from both sides. That’s why the two-sided limit fails to exist.