The expression $\lim_{x \to a} f(x) = \infty$ indicates how the function behaves as $x$ approaches $a$, not necessarily at $a$ itself. Let's evaluate each of the provided statements:

1. The limit exists and equals infinity
   True – The statement $\lim_{x \to a} f(x) = \infty$ signifies that the limit exists, and as $x$ approaches $a$, the function $f(x)$ increases without bound, which is interpreted as the limit "being" infinity.

2. The value of f(x) becomes infinite when x approaches a
   True – This is another way of saying that as $x$ gets closer to $a$, the values of $f(x)$ grow arbitrarily large (in other words, go to infinity).

3. The value of x goes out to infinity, when it gets close to a
   False – This statement is incorrect because it confuses the behavior of the function $f(x)$ with the behavior of $x$. As $x \to a$, the values of $x$ approach $a$, not infinity. It is the function $f(x)$, not $x$, that tends towards infinity.

4. As x approaches a, the value of f(x) increases without bound
   True – This is a correct interpretation. As $x$ gets closer to $a$, the function values increase indefinitely.

5. This means that f(a) = ∞
   False – The statement $\lim_{x \to a} f(x) = \infty$ says nothing about the value of $f(a)$. The function might not even be defined at $a$, and even if it is, the value of $f(a)$ could be finite or undefined. The limit describes the behavior near $a$, not at $a$.

Summary:

• True: Statements 1, 2, 4
• False: Statements 3, 5

Would you like more details or have any questions?

Related Questions:

1. What does it mean for a function to have a finite limit as $x \to a$?
2. How does $\lim_{x \to a^-} f(x) = \infty$ differ from $\lim_{x \to a^+} f(x) = \infty$?
3. Can a function be defined at $a$ if $\lim_{x \to a} f(x) = \infty$?
4. What is the difference between limits at infinity and limits approaching a finite value?
5. How does one evaluate limits involving infinity in rational functions?

Tip: Always remember that limits describe the behavior of a function near a point, not necessarily at the point itself.