Question

question 12 (1 point)

which of the following must be equal to \\(\lim_{x \to 3^-} f(x)\\) so that the limit exists at the point with the x-coordinate 3?

\\(f(3)\\)

\\(\lim_{x \to 3^+} f(x)\\)

nothing else. the one limit is sufficient.

0

Explanation:

Define the condition for limit existence

To understand when a limit exists at a specific point, we look at how the function behaves as it approaches that point from both sides. Using the Limit Evaluation concept, a two-sided limit \(\lim_{x \to c} f(x)\) exists if and only if both the left-sided limit and the right-sided limit exist and are equal to the same real number.

Apply the definition to the given point

We are given the left-sided limit:

$$\lim_{x \to 3^-} f(x)$$

For the two-sided limit to exist at the point with \(x\)-coordinate \(3\), this left-sided limit must equal the right-sided limit. Using the Limits of Piecewise Functions concept, we evaluate the behavior from the right:

$$\lim_{x \to 3^+} f(x)$$

Therefore, we must have:

$$\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x)$$

Evaluate the given options

Let's analyze the choices:

• \(f(3)\): This is the function value at \(x = 3\). The existence of a limit does not depend on the function being defined or continuous at that point.
• \(\lim_{x \to 3^+} f(x)\): This is the right-sided limit, which must equal the left-sided limit for the overall limit to exist.
• "Nothing else. The one limit is sufficient.": A single one-sided limit is not sufficient to guarantee the existence of a two-sided limit.
• \(0\): The limit does not have to equal \(0\); it can be any real number as long as both sides match.

Thus, the correct option is \(\lim_{x \to 3^+} f(x)\).

Answer:

• (A) \(f(3)\)
• (B) \(\lim_{x \to 3^+} f(x)\) (Correct answer)
• (C) Nothing else. The one limit is sufficient.
• (D) 0