If left and right limits exist and are equal to the same finite L, then the two-sided limit lim_{x->a} f(x) equals L.

If left and right limits exist and are equal to the same finite L, then the two-sided limit lim_{x->a} f(x) equals L.

• A. The limit exists only if the function is continuous at a.
• B. The limit exists only if the function is defined at a.
• C. If left and right limits exist and are equal to the same finite L, then the two-sided limit equals L.
• D. If left and right limits exist but are not equal, the limit still equals L.

Answer: (C)

When approaching a from both sides, you get a two-sided limit only if both one-sided limits exist and agree on the same value. If the left-hand limit exists and equals L, and the right-hand limit also exists and equals L, then approaching from either side brings you arbitrarily close to L, so the two-sided limit exists and equals L. The function’s value at a isn’t part of the limit, only the behavior near a matters. If the one-sided limits exist but are different, you would approach different values depending on the direction, so the two-sided limit does not exist. If one of the one-sided limits doesn’t exist, the two-sided limit can’t exist either. The finiteness condition just ensures we’re talking about a specific number L.

When approaching a from both sides, you get a two-sided limit only if both one-sided limits exist and agree on the same value. If the left-hand limit exists and equals L, and the right-hand limit also exists and equals L, then approaching from either side brings you arbitrarily close to L, so the two-sided limit exists and equals L. The function’s value at a isn’t part of the limit, only the behavior near a matters. If the one-sided limits exist but are different, you would approach different values depending on the direction, so the two-sided limit does not exist. If one of the one-sided limits doesn’t exist, the two-sided limit can’t exist either. The finiteness condition just ensures we’re talking about a specific number L.