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A (two-sided) limit exists even when the limit from the left and the limit from the right are nonexistent or not equal

a)True
b)False

User Slackware
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Final answer:

The statement is false. A two-sided limit can only exist if both the left-hand limit and the right-hand limit exist and are equal, otherwise, the two-sided limit does not exist.

Step-by-step explanation:

Regarding whether a two-sided limit exists when the limits from the left and the limit from the right are nonexistent or not equal, the answer is false. For a two-sided limit to exist at a certain point, the limit as the function approaches that point from the left (left-hand limit) and the limit as the function approaches from the right (right-hand limit) must both exist and be equal. If either the left-hand limit or the right-hand limit does not exist, or if they are not equal, then the two-sided limit at that point does not exist.

As an example, consider the function f(x) = 1/x. The two-sided limit as x approaches 0 does not exist because the function approaches positive infinity from the right and negative infinity from the left, resulting in different one-sided limits. Since the one-sided limits are not equal, the two-sided limit cannot exist.

User Mohamed Nageh
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