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

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 \( \lim_{x \rightarrow a} f(x) = \infty \).