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You have provided a list of exercises asking to find the derivative of the function using the limit process. Let's go through some of these problems step-by-step using the limit definition of a derivative:

### The limit definition of the derivative:
\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

#### Let's solve a few of these:
### 11. \( f(x) = 4 \)
This is a constant function. The derivative of any constant is 0.
\[
f'(x) = 0
\]

### 13. \( f(x) = -5x \)
This is a linear function. The derivative of a linear function \( ax \) is just the coefficient \( a \).
\[
f'(x) = -5
\]

### 15. \( f(x) = x^2 - 3 \)
To apply the limit definition:
\[
f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - 3 - (x^2 - 3)}{h}
\]
Expanding and simplifying:
\[
f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 3 - x^2 + 3}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h}
\]
\[
f'(x) = \lim_{h \to 0} (2x + h) = 2x
\]

### 17. \( f(x) = x^3 - 12x \)
Using the limit definition:
\[
f'(x) = \lim_{h \to 0} \frac{(x+h)^3 - 12(x+h) - (x^3 - 12x)}{h}
\]
Expanding:
\[
f'(x) = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - 12x - 12h - x^3 + 12x}{h}
\]
Simplifying:
\[
f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3 - 12h}{h} = \lim_{h \to 0} (3x^2 + 3xh + h^2 - 12)
\]
\[
f'(x) = 3x^2 - 12
\]

Would you like me to solve more of these exercises? Let me know if you'd like further details or explanations.

### Related Questions:
1. How does the limit definition of the derivative work for a linear function like \( f(x) = ax + b \)?
2. What is the derivative of \( f(x) = x^n \) for a general exponent \( n \)?
3. Why does the derivative of a constant function always equal zero?
4. Can you apply the limit process for functions involving products, like \( f(x) = x^2 \cdot g(x) \)?
5. How do higher-order derivatives relate to the limit process?

### Tip:
When computing derivatives using the limit definition, it's often helpful to expand all terms fully and cancel out similar ones before taking the limit.

Find the derivative of the function by the limit process in exercises 11-20.

This problem asks students to find the derivative of various functions using the limit process. Learn how to compute the derivative step-by-step for constant, linear, and polynomial functions. Appropriate for high school calculus students.

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