To find the derivative of the function $f(x) = x^2 - 3$ using the limit definition of a derivative, we will apply the formula:

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

Step-by-step process:

1. Substitute $f(x) = x^2 - 3$ into the limit definition:

$f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - 3 - (x^2 - 3)}{h}$

2. Expand $(x+h)^2$:

$(x+h)^2 = x^2 + 2xh + h^2$

Now substitute this into the equation:

$f'(x) = \lim_{h \to 0} \frac{(x^2 + 2xh + h^2 - 3) - (x^2 - 3)}{h}$

3. Simplify the expression:

First, cancel out common terms:

$f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 3 - x^2 + 3}{h}$

This simplifies to:

$f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h}$

4. Factor out $h$ from the numerator:

$f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h}$

Cancel the $h$:

$f'(x) = \lim_{h \to 0} (2x + h)$

5. Evaluate the limit as $h \to 0$:

$f'(x) = 2x$

Thus, the derivative of $f(x) = x^2 - 3$ is:

$f'(x) = 2x$

Would you like more details or have any questions?

---

Here are 5 related questions:

1. How do you find the derivative of $f(x) = x^3$ using the limit process?
2. What is the derivative of $f(x) = \frac{1}{x}$ using the limit definition?
3. Can you explain the difference between the power rule and the limit process for finding derivatives?
4. What does the derivative of a constant function look like?
5. How does the derivative of a quadratic function relate to its graph?

Tip: The limit definition of a derivative provides an intuitive understanding of how instantaneous rates of change are calculated.