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.

Explanation:

• The limit $\lim_{x

ightarrow a}f(x)=\infty$ is about the behavior of the function $f(x)$ as $x$ approaches $a$, not the value of the function at $x = a$. So, $f(a)=\infty$ is incorrect.

• The variable $x$ approaches $a$, not going to infinity. So, the statement about $x$ going to infinity is wrong.
• By the definition of the infinite limit $\lim_{x

ightarrow a}f(x)=\infty$, it means that as $x$ gets arbitrarily close to $a$ (from both sides), the values of $f(x)$ become arbitrarily large (increase without bound). And in the context of limits, we say the limit exists in the extended - real - number system when it is $\infty$ or $-\infty$.

Answer:

• 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.