Which of the following is/are signified by lim x->a f(x)=inf. 
 
 
-The limit exists and equals infinity. 
-The value of f(x) becomes infinite when x approaches a. 
-The value of x goes out to infinity, when it gets close to a. 
-As x approaches, the value of f(x) increases without bound. 
-This means that f(a) = inf.

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.

Which of the following is/are signified by lim x->a f(x)=inf?
-The limit exists and equals infinity.
-The value of f(x) becomes infinite when x approaches a.
-The value of x goes out to infinity, when it gets close to a.
-As x approaches, the value of f(x) increases without bound.
-This means that f(a) = inf.

Explore the concept of limits with the equation lim x->a f(x) = ∞. Learn how a function behaves as x approaches a, the significance of limits tending towards infinity, and why the value at f(a) might not be defined. Suitable for high school students in Grades 11-12.

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