If lim(x→3⁻) f(x) = 5 and lim(x→3⁺) f(x) = 7, what can be concluded?

Question

Accepted Answer

lim(x→3) f(x) does not exist. The correct answer is that the limit as x approaches 3 does not exist. For a two-sided limit to exist, both one-sided limits must exist and be equal. Since the left-hand limit (5) does not equal the right-hand limit (7), the two-sided limit does not exist. This situation indicates a jump discontinuity at x = 3.

Claiming the limit is 5 only considers the left-hand limit and ignores the right-hand limit. The two-sided limit cannot equal just one of the one-sided limits when they disagree. Claiming the limit is 6 (the average of 5 and 7) is incorrect; limits are not computed by averaging one-sided limits. Claiming the limit is 7 only considers the right-hand limit and ignores the left-hand limit.

One-sided limits and their relationship to two-sided limits are fundamental concepts tested on the CLEP exam. Students must understand that lim(x→a) f(x) = L if and only if both lim(x→a⁻) f(x) = L and lim(x→a⁺) f(x) = L. Jump discontinuities, where one-sided limits exist but differ, are the most common scenario where the two-sided limit fails to exist.

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