To solve this problem, we will use the limit definition of the derivative to find the slope of the tangent line to the curve $y = \sqrt{x} + 3$ at the point $(1, 2)$.

Limit Definition of Derivative:

The limit definition of the derivative is given by: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ Where $f(x) = \sqrt{x} + 3$.

Step-by-Step Solution:

1. Function at $x = 1$: $f(1) = \sqrt{1} + 3 = 1 + 3 = 4$

2. Function at $x + h$: $f(1 + h) = \sqrt{1 + h} + 3$

3. Substitute into the limit definition: $f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h} = \lim_{h \to 0} \frac{\left( \sqrt{1+h} + 3 \right) - 4}{h}$ Simplifying the numerator: $f'(1) = \lim_{h \to 0} \frac{\sqrt{1+h} + 3 - 4}{h} = \lim_{h \to 0} \frac{\sqrt{1+h} - 1}{h}$

4. Rationalize the numerator: Multiply both the numerator and the denominator by the conjugate $\sqrt{1+h} + 1$: $f'(1) = \lim_{h \to 0} \frac{\left( \sqrt{1+h} - 1 \right) \cdot \left( \sqrt{1+h} + 1 \right)}{h \cdot \left( \sqrt{1+h} + 1 \right)}$ Using the difference of squares, $(a - b)(a + b) = a^2 - b^2$, we get: $f'(1) = \lim_{h \to 0} \frac{(1 + h) - 1^2}{h \cdot (\sqrt{1+h} + 1)} = \lim_{h \to 0} \frac{h}{h \cdot (\sqrt{1+h} + 1)}$

5. Simplify: $f'(1) = \lim_{h \to 0} \frac{1}{\sqrt{1+h} + 1}$

6. Take the limit as $h \to 0$: As $h \to 0$, $\sqrt{1+h} \to 1$, so: $f'(1) = \frac{1}{1 + 1} = \frac{1}{2}$

Conclusion:

The slope of the tangent line to the curve $y = \sqrt{x} + 3$ at the point $(1, 2)$ is $\frac{1}{2}$.

Would you like more details on any specific part of this solution?

Here are 5 related questions:

1. What is the geometric interpretation of the derivative at a point?
2. How can you verify the slope of a tangent using the chain rule instead of the limit definition?
3. How does the process change if the function has a higher degree or is more complex?
4. Why is it necessary to rationalize the numerator when using the limit definition of the derivative?
5. How does the derivative of $\sqrt{x} + 3$ change as you move away from $x = 1$?

Tip: Rationalizing the numerator is a common technique when dealing with square roots in limits.