Let's analyze the problem from the image:

Problem:

Let $f$ be a function of one real variable. Suppose that:

$\lim_{x \to 4} f(x) = 3.$

You are asked to determine whether the following statements are True or False:

1. 3 is in the range/image of $f$.
2. $\lim_{x \to 4} (f(x))^2 = 9$.
3. 4 is in the domain of $f$.

Explanation:

1. 3 is in the range/image of $f$:

   • The fact that $\lim_{x \to 4} f(x) = 3$ does not guarantee that $f(4) = 3$ or that 3 is in the image of $f$. A limit describes the behavior of the function near a point, but $f(4)$ could be undefined or have a different value.
   • Answer: False.

2. $\lim_{x \to 4} (f(x))^2 = 9$:

   • If $\lim_{x \to 4} f(x) = 3$, squaring the limit gives $\lim_{x \to 4} (f(x))^2 = 3^2 = 9$.
   • Answer: True.

3. 4 is in the domain of $f$:

   • The limit $\lim_{x \to 4} f(x) = 3$ implies that $f(x)$ is defined and behaves well as $x \to 4$, but this does not mean that $f(4)$ is defined. There are cases where a limit exists, but the function is not defined at the point itself (e.g., removable discontinuity).
   • Answer: False.

Recap:

• Statement 1: False.
• Statement 2: True.
• Statement 3: False.

Would you like any further details or clarifications?

Related questions:

1. What is the formal definition of a limit?
2. How do limits relate to continuity of a function?
3. Can a function have a limit at a point where it is not defined?
4. What are examples of limits that do not exist?
5. How does squaring a function affect its limit?

Tip:

Always remember, a limit describes the behavior near a point but does not necessarily reflect the value of the function at that point!