Which of the following is/are signified by ?
The value of goes out to infinity, when it gets close to .
This means that
As approaches , the value of increases without bound.
The limit exists and equals infinity.
The value of becomes infinite when approaches .

Ask by Hill Nichols. in the United States

Jan 27,2025

Question

Which of the following is/are signified by ?
The value of goes out to infinity, when it gets close to .
This means that
As approaches , the value of increases without bound.
The limit exists and equals infinity.
The value of becomes infinite when approaches .

Ask by Hill Nichols. in the United States

Jan 27,2025

UpStudy AI · Accepted Answer

The expression $$ \lim_{x \rightarrow a} f(x) = \infty $$ conveys specific information about the behavior of the function $$ f(x) $$ as $$ x $$ approaches the value $$ a $$. Let's evaluate each of the provided statements to determine which are accurate interpretations of this limit notation.

1. **□ The value of $$ x $$ goes out to infinity, when it gets close to $$ a $$.**

**Incorrect.**  
   The notation $$ \lim_{x \rightarrow a} f(x) = \infty $$ describes the behavior of $$ f(x) $$ as $$ x $$ approaches $$ a $$. It does **not** imply anything about $$ x $$ itself going to infinity.

2. **□ This means that $$ f(a) = \infty $$.**

**Incorrect.**  
   The limit describes the behavior of $$ f(x) $$ as $$ x $$ approaches $$ a $$, but it does not necessarily define the value of $$ f $$ at $$ a $$. In many cases, $$ f(a) $$ might be undefined or not equal to infinity, even though the limit is infinite.

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

**Correct.**  
   This is a direct and accurate interpretation of $$ \lim_{x \rightarrow a} f(x) = \infty $$. It means that as $$ x $$ gets closer to $$ a $$, $$ f(x) $$ grows larger and larger without any upper limit.

4. **□ The limit exists and equals infinity.**

**Conditionally Correct.**  
   In the extended real number system, infinity is considered a possible limit, so this statement is **correct** within that context. However, in the strict real number system, limits are required to approach finite values, and infinity is not considered a real number. Typically, in calculus and most mathematical contexts, saying a limit "equals infinity" is acceptable and understood to mean that the function grows without bound.

5. **□ The value of $$ f(x) $$ becomes infinite when $$ x $$ approaches $$ a $$.**

**Correct.**  
   This is another accurate interpretation of the limit notation. It succinctly states that $$ f(x) $$ grows beyond all finite bounds as $$ x $$ nears $$ a $$.

### **Summary of Correct Statements**

- **As $$ x $$ approaches $$ a $$, the value of $$ f(x) $$ increases without bound.**
- **The limit exists and equals infinity.** *(Within the extended real number system)*
- **The value of $$ f(x) $$ becomes infinite when $$ x $$ approaches $$ a $$.**

These statements correctly represent the meaning of $$ \lim_{x \rightarrow a} f(x) = \infty $$.
As $$ x $$ approaches $$ a $$, $$ f(x) $$ increases without bound.

Which of the following is/are signified by ?
The value of goes out to infinity, when it gets close to .
This means that
As approaches , the value of increases without bound.
The limit exists and equals infinity.
The value of becomes infinite when approaches .

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Which of the following is/are signified by ? The value of goes out to infinity, when it gets close to . This means that As approaches , the value of increases without bound. The limit exists and equals infinity. The value of becomes infinite when approaches .

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Which of the following is/are signified by ?
The value of goes out to infinity, when it gets close to .
This means that
As approaches , the value of increases without bound.
The limit exists and equals infinity.
The value of becomes infinite when approaches .