Questions: Which of the following is/are signified by lim x → a f(x)=∞ ? This means that f(a)=∞ The limit exists and equals infinity. As x approaches a, the value of f(x) increases without bound. 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.

Solution Steps

To determine which statements are signified by \(\lim _{x \rightarrow a} f(x)=\infty\), we need to understand the definition of a limit approaching infinity. Specifically, this means that as \(x\) gets arbitrarily close to \(a\), the value of \(f(x)\) increases without bound. We will evaluate each statement to see if it aligns with this definition.

The expression \(\lim _{x \rightarrow a} f(x)=\infty\) means that as \(x\) approaches \(a\), the value of \(f(x)\) increases without bound. This does not necessarily mean that \(f(a)\) is defined or that \(f(a) = \infty\).

We will evaluate each statement based on the definition of the limit:

1. This means that \(f(a)=\infty\): This is false because the limit approaching infinity does not imply that the function value at \(a\) is infinite or even defined.
2. The limit exists and equals infinity: This is true because the limit definition states that as \(x\) approaches \(a\), \(f(x)\) increases without bound.
3. As \(x\) approaches \(a\), the value of \(f(x)\) increases without bound: This is true and directly aligns with the definition of the limit.
4. The value of \(x\) goes out to infinity, when it gets close to \(a\): This is false because it is the value of \(f(x)\) that goes to infinity, not \(x\).
5. The value of \(f(x)\) becomes infinite when \(x\) approaches \(a\): This is true and restates the definition of the limit.

Final Answer