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

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

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

Solution

Solution Steps

To determine which statements are signified by $\lim _{x \rightarrow a} f(x)=\infty$, we need to understand the definition of this limit. Specifically, it means that as $x$ approaches $a$, the function $f(x)$ increases without bound. This does not necessarily mean that $f(a)$ is defined or that the limit exists in the traditional sense (as a finite number).

Solution Approach

The statement "This means that $f(a)=\infty$" is incorrect because the limit describes the behavior as $x$ approaches $a$, not the value at $a$.
The statement "The value of $x$ goes out to infinity, when it gets close to $a$" is incorrect because it is $f(x)$ that goes to infinity, not $x$.
The statement "As $x$ approaches $a$, the value of $f(x)$ increases without bound" is correct.
The statement "The limit exists and equals infinity" is correct in the context of limits approaching infinity.
The statement "The value of $f_{l}(x)$ becomes infinite when $x$ approaches $a$" is correct if $f_{l}(x)$ is interpreted as $f(x)$.

Step 1: Analyze the Statements

We need to determine which statements are signified by $\lim _{x \rightarrow a} f(x)=\infty$. This limit means that as $x$ approaches $a$, the function $f(x)$ increases without bound.

Step 2: Evaluate Each Statement

This means that $f(a)=\infty$: This is incorrect because the limit describes the behavior as $x$ approaches $a$, not the value at $a$.
The value of $x$ goes out to infinity, when it gets close to $a$: This is incorrect because it is $f(x)$ that goes to infinity, not $x$.
As $x$ approaches $a$, the value of $f(x)$ increases without bound: This is correct.
The limit exists and equals infinity: This is correct in the context of limits approaching infinity.
The value of $f_{l}(x)$ becomes infinite when $x$ approaches $a$: This is correct if $f_{l}(x)$ is interpreted as $f(x)$.

Final Answer

The correct statements are:

As $x$ approaches $a$, the value of $f(x)$ increases without bound.
The limit exists and equals infinity.
The value of $f_{l}(x)$ becomes infinite when $x$ approaches $a$.

\[ \boxed{\text{As } x \text{ approaches } a, \text{ the value of } f(x) \text{ increases without bound.}} \] \[ \boxed{\text{The limit exists and equals infinity.}} \] \[ \boxed{\text{The value of } f_{l}(x) \text{ becomes infinite when } x \text{ approaches } a.} \]

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