Question

which of the following is/are signified by \\( \lim _ { x \
ightarrow a } f ( x ) = \infty \\)?
this means that \\( f ( a ) = \infty \\)
the value of \\( x \\) goes out to infinity, when it gets close to \\( a \\).
the value of \\( f ( x ) \\) becomes infinite when \\( x \\) approaches \\( a \\).
the limit exists and equals infinity.
as \\( x \\) approaches \\( a \\), the value of \\( f ( x ) \\) increases without bound.
question help: message instructor

Explanation:

Step1: Analyze the first option

The limit $\lim_{x
ightarrow a}f(x)=\infty$ is about the behavior of $f(x)$ as $x$ approaches $a$, not the value of $f(a)$. $f(a)$ may not even be defined. So, "This means that $f(a)=\infty$" is incorrect.

Step2: Analyze the second option

The limit $\lim_{x
ightarrow a}f(x)=\infty$ is about the behavior of $f(x)$ as $x$ approaches $a$, not the value of $x$. $x$ approaches $a$, not infinity. So, "The value of $x$ goes out to infinity, when it gets close to $a$" is incorrect.

Step3: Analyze the third option

By the definition of $\lim_{x
ightarrow a}f(x)=\infty$, it means that the value of $f(x)$ becomes infinite (increases without bound) when $x$ approaches $a$. So, "The value of $f(x)$ becomes infinite when $x$ approaches $a$" is correct.

Step4: Analyze the fourth option

In the context of real - valued functions (in the basic calculus sense), when we write $\lim_{x
ightarrow a}f(x)=\infty$, we are using the extended real - number system notation. Formally, in the real - number system (without the extended real numbers), the limit does not exist in the traditional sense (since infinity is not a real number). But in the extended real - number system (where we add $\pm\infty$ to the real numbers), we can say the limit exists and equals infinity.

Step5: Analyze the fifth option

The definition of $\lim_{x
ightarrow a}f(x)=\infty$ is that for any $M>0$, there exists a $\delta>0$ such that if $0 < |x - a|<\delta$, then $f(x)>M$. This means as $x$ approaches $a$, the value of $f(x)$ increases without bound. So, "As $x$ approaches $a$, the value of $f(x)$ increases without bound" is correct.

Answer:

C. The value of \( f(x) \) becomes infinite when \( x \) approaches \( a \).
D. The limit exists and equals infinity.
E. As \( x \) approaches \( a \), the value of \( f(x) \) increases without bound.