find g'(x) by using limit process g(x)=√x-2

To find \( g'(x) \) using the limit definition of a derivative for the function \( g(x) = \sqrt{x} - 2 \), we can follow these steps:

The limit definition of a derivative is:

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

### Step 1: Substitute the function \( g(x) = \sqrt{x} - 2 \) into the limit definition.

\[
g'(x) = \lim_{{h \to 0}} \frac{\left( \sqrt{x+h} - 2 \right) - \left( \sqrt{x} - 2 \right)}{h}
\]

### Step 2: Simplify the numerator.

\[
g'(x) = \lim_{{h \to 0}} \frac{\sqrt{x+h} - \sqrt{x}}{h}
\]

### Step 3: Multiply the numerator and denominator by the conjugate of the numerator.

\[
g'(x) = \lim_{{h \to 0}} \frac{\left( \sqrt{x+h} - \sqrt{x} \right) \cdot \left( \sqrt{x+h} + \sqrt{x} \right)}{h \cdot \left( \sqrt{x+h} + \sqrt{x} \right)}
\]

### Step 4: Apply the difference of squares in the numerator.

\[
g'(x) = \lim_{{h \to 0}} \frac{(x+h) - x}{h \cdot \left( \sqrt{x+h} + \sqrt{x} \right)}
\]

\[
g'(x) = \lim_{{h \to 0}} \frac{h}{h \cdot \left( \sqrt{x+h} + \sqrt{x} \right)}
\]

### Step 5: Cancel out \( h \) in the numerator and denominator.

\[
g'(x) = \lim_{{h \to 0}} \frac{1}{\sqrt{x+h} + \sqrt{x}}
\]

### Step 6: Apply the limit as \( h \to 0 \).

\[
g'(x) = \frac{1}{\sqrt{x} + \sqrt{x}} = \frac{1}{2\sqrt{x}}
\]

### Final Answer:
\[
g'(x) = \frac{1}{2\sqrt{x}}
\]

Let me know if you need more details or have any questions!

Here are five related questions to explore further:
1. How can we apply the limit process to differentiate other radical functions?
2. What does the derivative of \( g(x) = \sqrt{x} - 2 \) represent graphically?
3. Can you find the second derivative \( g''(x) \) of the given function?
4. How would we differentiate \( g(x) = \sqrt{x+1} - 2 \) using the limit process?
5. How can we interpret the behavior of \( g'(x) \) as \( x \to 0 \) and \( x \to \infty \)?

**Tip**: When dealing with square roots, multiplying by the conjugate is a powerful technique to simplify expressions, especially when using limits.

Explore the step-by-step process to find the derivative of g(x) = √x - 2 using the limit definition of a derivative. Learn key calculus concepts, such as the limit process and difference of squares, suitable for Grades 11-12 and early college students.

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