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The problem shown in the image is asking to find the derivative of a function using the definition of the derivative, which is given by the following limit:

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

This is the standard definition of the derivative of a function \( f(x) \).

In the specific part (a), the function is:

\[
f(x) = 2x + 3
\]

Let's calculate the derivative using the definition of the limit:

1. **Substitute the function \( f(x) \) into the definition:**

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

   First, find \( f(x+h) \):
   \[
   f(x+h) = 2(x+h) + 3 = 2x + 2h + 3
   \]

2. **Substitute \( f(x+h) \) and \( f(x) \) into the limit:**

   \[
   \lim_{h \to 0} \frac{(2x + 2h + 3) - (2x + 3)}{h}
   \]

   Simplify the numerator:
   \[
   (2x + 2h + 3) - (2x + 3) = 2h
   \]

3. **Simplify the fraction:**

   \[
   \frac{2h}{h} = 2
   \]

4. **Take the limit as \( h \to 0 \):**

   \[
   \lim_{h \to 0} 2 = 2
   \]

So, the derivative of \( f(x) = 2x + 3 \) is:

\[
f'(x) = 2
\]

This matches the result you would expect since the derivative of a linear function \( 2x + 3 \) is the slope of the line, which is constant at 2.

Would you like further details, or do you have any other questions?

Here are 5 related questions you might find helpful:
1. How does the definition of the derivative relate to the slope of a function at a point?
2. What is the derivative of a constant, and why?
3. Can the derivative of a polynomial function be found using the same limit definition?
4. How would you differentiate a quadratic function like \( f(x) = x^2 + 2x + 1 \)?
5. Why does the derivative of \( f(x) = 2x + 3 \) not depend on \( x \)?

**Tip:** When working with derivatives, recognizing common forms (like polynomials or trigonometric functions) can simplify calculations significantly.

Encuentre el límite lim (h -> 0) [f(x+h) - f(x)] / h para f(x) = 2x + 3.

Find the derivative of the linear function f(x) = 2x + 3 using the definition of the derivative: lim (h -> 0) [f(x+h) - f(x)] / h. This problem covers fundamental concepts of calculus, including limits and the derivative of a linear function.

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