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The problem asks to:

1. Use the limit definition to compute the derivative of the function \( f(x) = \sqrt{x} + 12 \) at \( x = 1 \).
2. Find the equation of the tangent line to the graph of \( f \) at \( x = 1 \).

### Step 1: Using the Limit Definition of the Derivative

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

For \( f(x) = \sqrt{x} + 12 \), let's compute the derivative at \( x = 1 \).

\[
f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h}
\]

Now, calculate each part:
- \( f(1) = \sqrt{1} + 12 = 1 + 12 = 13 \)
- \( f(1 + h) = \sqrt{1 + h} + 12 \)

Substitute into the limit definition:
\[
f'(1) = \lim_{h \to 0} \frac{\left( \sqrt{1+h} + 12 \right) - 13}{h}
\]
\[
f'(1) = \lim_{h \to 0} \frac{\sqrt{1+h} - 1}{h}
\]

To simplify this, multiply the numerator and denominator by the conjugate of the numerator:
\[
f'(1) = \lim_{h \to 0} \frac{\sqrt{1+h} - 1}{h} \cdot \frac{\sqrt{1+h} + 1}{\sqrt{1+h} + 1}
\]
\[
f'(1) = \lim_{h \to 0} \frac{(1+h) - 1^2}{h(\sqrt{1+h} + 1)} = \lim_{h \to 0} \frac{h}{h(\sqrt{1+h} + 1)}
\]
\[
f'(1) = \lim_{h \to 0} \frac{1}{\sqrt{1+h} + 1}
\]

As \( h \to 0 \), \( \sqrt{1+h} \to 1 \). Therefore:
\[
f'(1) = \frac{1}{1 + 1} = \frac{1}{2}
\]

So, \( f'(1) = \frac{1}{2} \).

### Step 2: Equation of the Tangent Line

The equation of the tangent line is given by:
\[
y - f(1) = f'(1)(x - 1)
\]

We already know that:
- \( f'(1) = \frac{1}{2} \)
- \( f(1) = 13 \)

Substitute into the tangent line equation:
\[
y - 13 = \frac{1}{2}(x - 1)
\]

Simplify:
\[
y = \frac{1}{2}(x - 1) + 13
\]
\[
y = \frac{1}{2}x - \frac{1}{2} + 13
\]
\[
y = \frac{1}{2}x + \frac{25}{2}
\]

Thus, the equation of the tangent line is:
\[
y = \frac{1}{2}x + \frac{25}{2}
\]

---

Would you like further details on any steps? Here are some related questions:

1. How is the conjugate used to simplify expressions involving square roots?
2. What does the limit definition of the derivative represent geometrically?
3. Why is the derivative of a constant like \( 12 \) equal to zero?
4. How can we generalize this process for any square root function?
5. Can you compute the second derivative of \( f(x) = \sqrt{x} + 12 \)?

**Tip:** When simplifying limits involving square roots, multiplying by the conjugate is a powerful technique to eliminate radicals.

Use the limit definition to compute the derivative of the function f(x) = √x + 12 at x = 1. Find an equation of the tangent line to the graph of f at x = 1.

Learn how to compute the derivative of f(x) = √x + 12 at x = 1 using the limit definition and find the equation of the tangent line. This problem involves fundamental calculus concepts, such as the limit definition of a derivative and equations of tangent lines.

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