Let's break down the problem step by step, starting with each part separately.

1. Evaluate the limit expression:

The limit expression given is:

$\lim_{h \to 0} \frac{f(3+h) - f(3)}{h}$

This limit represents the derivative of the function $f(x)$ at $x = 3$. Specifically, it is:

$f'(3) = \lim_{h \to 0} \frac{f(3+h) - f(3)}{h}$

To determine whether this limit exists, we need more information about the function $f(x)$. If $f(x)$ is differentiable at $x = 3$, then this limit exists and equals $f'(3)$.

2. Does the limit as $x$ approaches 5 exist?

The question asks whether $\lim_{x \to 5} f(x)$ exists. For this limit to exist, the function $f(x)$ must approach a specific value as $x$ gets arbitrarily close to 5 from both sides (left and right). Again, this depends on the behavior of $f(x)$ near $x = 5$.

3. Is $f(x)$ continuous on the closed interval $[-1, 7]$?

A function $f(x)$ is continuous on a closed interval $[-1, 7]$ if:

• $f(x)$ is continuous at every point in the open interval $(-1, 7)$.
• $f(x)$ is continuous from the right at $x = -1$.
• $f(x)$ is continuous from the left at $x = 7$.

To determine continuity, we need to check if the limits $\lim_{x \to c} f(x)$ exist and equal $f(c)$ for all $c$ in $[-1, 7]$.

4. Is there a $c$ in $(0, 7)$ such that $f(c) = -2$?

This part of the problem is related to the Intermediate Value Theorem (IVT). The IVT states that if $f(x)$ is continuous on a closed interval $[a, b]$, and $N$ is any number between $f(a)$ and $f(b)$, then there exists at least one number $c$ in the open interval $(a, b)$ such that $f(c) = N$.

To answer whether there exists some $c$ in $(0, 7)$ such that $f(c) = -2$, we need to know:

• If $f(x)$ is continuous on $[0, 7]$.
• The values $f(0)$ and $f(7)$.

If $f(x)$ is continuous on $[0, 7]$ and $-2$ is between $f(0)$ and $f(7)$, then by the IVT, there exists some $c$ in $(0, 7)$ such that $f(c) = -2$.

Summary:

To fully answer the questions posed, we need more specific information about the function $f(x)$. If provided, we can determine:

1. Whether $f'(3)$ exists (the derivative at $x = 3$).
2. Whether the limit $\lim_{x \to 5} f(x)$ exists.
3. Whether $f(x)$ is continuous on the interval $[-1, 7]$.
4. Whether there is a $c$ such that $f(c) = -2$ in the interval $(0, 7)$.

Would you like to provide more details on $f(x)$ or explore these concepts further?

Here are 5 related questions:

1. What are the conditions for a function to be differentiable at a point?
2. How do you determine if a limit exists at a particular point?
3. What is the significance of the Intermediate Value Theorem in calculus?
4. How can you check the continuity of a function on a closed interval?
5. What is the relationship between differentiability and continuity?

Tip: Always check the continuity of a function before applying the Intermediate Value Theorem.