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Let's go over the answers directly:

### (a) Simplify \( \frac{f(21 + h) - f(21)}{h} \)

Given \( f(x) = \sqrt{x} \):

1. \( f(21 + h) = \sqrt{21 + h} \)
2. \( f(21) = \sqrt{21} \)

Thus, the expression becomes:

\[
\frac{\sqrt{21 + h} - \sqrt{21}}{h}
\]

To simplify, multiply the numerator and denominator by the conjugate of the numerator:

\[
\frac{\left( \sqrt{21 + h} - \sqrt{21} \right)}{h} \cdot \frac{\left( \sqrt{21 + h} + \sqrt{21} \right)}{\left( \sqrt{21 + h} + \sqrt{21} \right)} = \frac{(21 + h) - 21}{h(\sqrt{21 + h} + \sqrt{21})}
\]

Simplifying further:

\[
\frac{h}{h(\sqrt{21 + h} + \sqrt{21})} = \frac{1}{\sqrt{21 + h} + \sqrt{21}}
\]

### (b) Evaluate \( \lim_{h \to 0} \frac{f(21 + h) - f(21)}{h} \)

Using the simplified form from part (a), we take the limit as \( h \to 0 \):

\[
\lim_{h \to 0} \frac{1}{\sqrt{21 + h} + \sqrt{21}} = \frac{1}{\sqrt{21} + \sqrt{21}} = \frac{1}{2\sqrt{21}}
\]

### (c) Equation of the tangent line at \( x = 21 \)

The slope of the tangent line is given by the derivative \( f'(x) \) evaluated at \( x = 21 \). From part (b), the derivative at \( x = 21 \) is:

\[
f'(21) = \frac{1}{2\sqrt{21}}
\]

The equation of the tangent line is given by:

\[
y - f(21) = f'(21)(x - 21)
\]

Since \( f(21) = \sqrt{21} \), the equation becomes:

\[
y - \sqrt{21} = \frac{1}{2\sqrt{21}}(x - 21)
\]

Let me know if you'd like further details or have any questions.

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Here are 5 related questions to extend understanding:
1. How do we apply the conjugate method to rationalize more complex expressions?
2. What are the geometric interpretations of limits in derivative calculations?
3. How does the concept of instantaneous rate of change apply in physics?
4. Why does the limit exist in this case, and what conditions might cause a limit to not exist?
5. How would the tangent line equation change if \( f(x) \) were a cubic function?

**Tip**: Always remember that the derivative \( f'(x) \) gives you the slope of the tangent line at any point \( x \).

Explore the solution to calculating the derivative and equation of the tangent line for the square root function at x=21 using limit definitions and algebraic simplifications. This guide covers calculus concepts suitable for college-level students.

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