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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.

To find the derivative of y = √x - 3 at x = 7 using derivative at a point, which of the following is correct?

Find the derivative of y = √x - 3 at x = 7 using the limit definition of the derivative. This solution covers the key steps in applying limits to find the derivative of square root functions, including simplifying the expression for accurate results.

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