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.