Certainly! Here are multiple-choice answers for the problem involving the limit $\lim_{x \to 2} f(x) = 5$:

Question:

Given that $\lim_{x \to 2} f(x) = 5$, which of the following statements must be true?

A) $f(2) = 5$

B) As $x$ approaches 2, the value of $f(x)$ approaches 5.

C) There exists a value $c$ near 2 such that $f(c) = 5$.

D) For every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - 2| < \delta$, then $|f(x) - 5| < \epsilon$.

E) $f(x)$ is continuous at $x = 2$.

Correct Answer(s):

• B: This is the definition of a limit. As $x$ approaches 2, $f(x)$ approaches 5.
• D: This is the precise definition of a limit. It states that for every small distance $\epsilon$ from 5, there is a corresponding distance $\delta$ from 2 such that $f(x)$ stays within $\epsilon$ of 5 whenever $x$ is within $\delta$ of 2.

Incorrect Answer(s) Explanation:

• A: This statement is not necessarily true. The limit of $f(x)$ as $x$ approaches 2 being 5 does not require that $f(2) = 5$.
• C: This statement suggests a specific value near 2 where $f(x) = 5$, but the limit only guarantees that $f(x)$ approaches 5 as $x$ approaches 2, not that it necessarily equals 5 at any specific point.
• E: Continuity at $x = 2$ would require $f(2) = 5$, which is not guaranteed by the limit alone.

Would you like more details or have any questions?

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Relative Questions:

1. What is the definition of the limit $\lim_{x \to a} f(x) = L$?
2. How does continuity at a point relate to the limit at that point?
3. Can the limit $\lim_{x \to a} f(x)$ exist if $f(x)$ has a discontinuity at $x = a$?
4. What is the difference between a limit existing and a function being continuous?
5. How can you determine if a limit does not exist?

Tip: When evaluating limits, remember that the value of the function at the limit point is irrelevant unless the problem explicitly asks about continuity.