To determine the derivative of $y = \sqrt{x} - 3$ at $x = 7$ using the definition of the derivative, we use the limit definition:

$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

For the given function $y = \sqrt{x} - 3$, the derivative at $x = 7$ becomes:

$f'(7) = \lim_{x \to 7} \frac{\sqrt{x} - 3 - (\sqrt{7} - 3)}{x - 7}$

Since $\sqrt{7} - 3$ is a specific numerical value, the expression simplifies to:

$f'(7) = \lim_{x \to 7} \frac{\sqrt{x} - 2}{x - 7}$

Thus, the correct answer is (c):

$\lim_{x \to 7} \frac{\sqrt{x} - 2}{x - 7}$

Would you like more details about how limits work for derivatives, or how to solve this step-by-step?

Here are 5 related questions to deepen your understanding:

1. How does the limit definition of a derivative work in general cases?
2. Why is $\sqrt{7} - 3$ simplified to $2$ in this example?
3. What happens if the function had a more complex expression, such as $y = x^2 + \sqrt{x} - 3$?
4. Can we compute the derivative of this function using differentiation rules directly without limits?
5. How does the concept of limits help in determining the slope of tangent lines?

Tip: When simplifying derivative problems involving square roots, rationalizing the numerator often helps clarify the limit calculation.