Question

Determine whether the statement is true or false, and explain why. For the limit of f(x) as x→a to exist, the limit from the left and the limit from the right must both exist and be the same. Choose the correct answer below.

A. This statement is true because it is part of the definition of the existence of a limit.
B. This statement is false because one of the limits could be infinity while the other is negative infinity, so while the one-sided limits exist, the two-sided limit does not exist.
C. This statement is false because the limit from the left and the limit from the right could both be infinity or both be negative infinity, so while the one-sided limits do not exist, the two-sided limit exists.
D. This statement is false because f(x) must also be defined at x=a.

Answer 1

The statement is true because it reflects the definition of a limit, which requires that the one-sided limits from both the left and the right approach the same value for the two-sided limit to exist.Option A is the correct answer.

Step-by-step explanation:

The statement, "For the limit of f(x) as x→a to exist, the limit from the left and the limit from the right must both exist and be the same," is true.

This is indeed part of the definition of the existence of a limit in mathematics. For a two-sided limit to exist, the one-sided limits must approach the same value as x approaches a from both directions. If the left-hand limit and the right-hand limit are not equal, then the two-sided limit does not exist.

This is also true regardless of whether the function is defined at x=a; the existence of a limit solely depends on the behavior of the function as it approaches the point from either side, not on the actual value of the function at the point.