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To solve this problem, we need to evaluate the expression:

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

This expression represents the derivative of \( f(x) \) at \( x = 3 \). From the graph, let’s analyze the values of \( f(3) \) and \( f(3 + h) \) as \( h \) approaches 0 to determine this derivative.

### Step-by-Step Solution
1. **Determine \( f(3) \):**
   Looking at the graph, the value of \( f(3) \) is at \( y = -1 \).

   \[
   f(3) = -1
   \]

2. **Determine \( f(3 + h) \) as \( h \to 0 \):**
   We need to look at the behavior of \( f(x) \) around \( x = 3 \). Observing the slope of the function near \( x = 3 \) on the graph, it appears that there is a constant slope as \( x \) increases from \( x = 2 \) to \( x = 4 \).

   From the graph, \( f(2) = -3 \) and \( f(4) = 0 \). So, the slope between these points is:

   \[
   \frac{{f(4) - f(2)}}{{4 - 2}} = \frac{0 - (-3)}{2} = \frac{3}{2} = 1.5
   \]

3. **Conclusion:**
   The derivative \( f'(3) \) is equal to the slope between these points, which is \( \frac{3}{2} \) or \( 1.5 \).

Thus,

\[
\lim_{{h \to 0}} \frac{{f(3 + h) - f(3)}}{h} = 1.5
\]

### Answer:
\[
\lim_{{h \to 0}} \frac{{f(3 + h) - f(3)}}{h} = 1.5
\]

Would you like further explanation, or do you have any questions about this solution?

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Here are five related questions to deepen your understanding:

1. How is the derivative defined in terms of limits?
2. How can we find the slope of a curve at a specific point using the graph?
3. What are the implications if the limit in this problem does not exist?
4. How does a graph help us interpret the derivative as a rate of change?
5. What if the function had a sharp corner at \( x = 3 \); how would this affect the derivative?

**Tip:** When evaluating derivatives graphically, always ensure that the function is continuous and differentiable at the point of interest. Discontinuities or sharp corners indicate that the derivative may not exist.

Evaluate \( \lim_{{h \to 0}} \frac{{f(3+h) - f(3)}}{h} \) using the provided graph of \( y = f(x) \).

Learn how to evaluate the derivative of f(x) at x = 3 using the limit \( \lim_{{h \to 0}} \frac{{f(3+h) - f(3)}}{h} \) with graphical interpretation. This problem focuses on limits, derivatives, and slope calculation from a graph, suitable for high school calculus students.

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