The image shows a question asking about the interpretation of the mathematical limit $\lim_{x \to a} f(x) = \infty$.

This expression means that as $x$ approaches $a$, the value of the function $f(x)$ grows without bound, essentially tending to infinity. Let's review each of the options:

1. "As $x$ approaches $a$, the value of $f(x)$ increases without bound."

   • Correct: This directly describes the behavior indicated by the limit.

2. "The value of $x$ goes out to infinity, when it gets close to $a$."

   • Incorrect: The limit is about $f(x)$, not $x$. This statement suggests that $x$ goes to infinity, which is not what the limit means.

3. "The value of $f(x)$ becomes infinite when $x$ approaches $a$."

   • Correct: This restates the definition of the limit going to infinity.

4. "This means that $f(a) = \infty$."

   • Incorrect: The limit as $x$ approaches $a$ does not necessarily mean $f(a)$ itself is infinite. The function may not even be defined at $a$.

5. "The limit exists and equals infinity."

   • Correct: A limit can exist and be infinite. This is the correct interpretation of a limit approaching infinity.

Summary:

The correct options are:

• Option 1: "As $x$ approaches $a$, the value of $f(x)$ increases without bound."
• Option 3: "The value of $f(x)$ becomes infinite when $x$ approaches $a$."
• Option 5: "The limit exists and equals infinity."

Let me know if you want further details or explanations.

Here are five related questions you could explore:

1. What is the difference between limits approaching a finite number and limits approaching infinity?
2. How does one handle limits where $f(x)$ tends to infinity from one side but not the other?
3. What happens when $\lim_{x \to a} f(x) = 0$?
4. Can limits exist for discontinuous functions at a point?
5. What is the relationship between limits and function continuity?

Tip: A limit approaching infinity means the function grows larger and larger, but this doesn't guarantee anything about the function's value at the specific point.