https://file.aiquickdraw.com/mathbot/file/17302238937976b3563oz.jpg

To solve this limit, let's recognize that it represents the derivative of \( f(x) = 4x - 3 \) at \( x = 3 \).

### Step-by-Step Solution:

1. **Substitute \( f(x) = 4x - 3 \) into the limit definition:**

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

2. **Calculate \( f(3+h) \) and \( f(3) \):**
   - \( f(3+h) = 4(3+h) - 3 = 12 + 4h - 3 = 9 + 4h \)
   - \( f(3) = 4 \cdot 3 - 3 = 12 - 3 = 9 \)

3. **Substitute these values into the limit expression:**

   \[
   \lim_{h \to 0} \frac{(9 + 4h) - 9}{h} = \lim_{h \to 0} \frac{4h}{h}
   \]

4. **Simplify the expression:**

   \[
   = \lim_{h \to 0} 4 = 4
   \]

### Conclusion:
The limit is \( \boxed{4} \).

This means the derivative of \( f(x) = 4x - 3 \) at \( x = 3 \) is 4.

Would you like more details, or do you have any questions?

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Here are some related questions for further exploration:

1. What is the general derivative of \( f(x) = 4x - 3 \)?
2. How would you find the slope of the tangent line at a different point, say \( x = 5 \)?
3. What does the derivative of \( f(x) = 4x - 3 \) tell us about the function's graph?
4. How does the concept of the derivative as a limit apply to non-linear functions?
5. What happens to the limit expression if \( f(x) \) were not a linear function?

**Tip:** For linear functions \( f(x) = ax + b \), the derivative is always the constant \( a \), as the slope is consistent across all points.

Find the limit for f(x) = 4x - 3: \( \lim_{{h \to 0}} \frac{{f(3 + h) - f(3)}}{h} \).

Learn how to calculate the derivative of f(x) = 4x - 3 at x = 3 using the limit definition. This solution covers key calculus concepts including limits and derivatives, suitable for high school students in Grades 11-12.

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