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We are tasked with solving the following limit:  

\[
\lim_{x \to 0} \frac{\sqrt{x+2} - \sqrt{2}}{x(1+x)}.
\]

---

### Step 1: Simplify the numerator  
The numerator involves a difference of square roots:  
\[
\sqrt{x+2} - \sqrt{2}.
\]  
To simplify, multiply and divide by the conjugate:  
\[
\frac{\sqrt{x+2} - \sqrt{2}}{\sqrt{x+2} - \sqrt{2}} \cdot \frac{\sqrt{x+2} + \sqrt{2}}{\sqrt{x+2} + \sqrt{2}} = \frac{(\sqrt{x+2})^2 - (\sqrt{2})^2}{\sqrt{x+2} + \sqrt{2}}.
\]

This simplifies to:  
\[
\frac{(x+2) - 2}{\sqrt{x+2} + \sqrt{2}} = \frac{x}{\sqrt{x+2} + \sqrt{2}}.
\]

Thus, the numerator becomes:  
\[
\sqrt{x+2} - \sqrt{2} = \frac{x}{\sqrt{x+2} + \sqrt{2}}.
\]

---

### Step 2: Rewrite the full expression  
Substitute the simplified numerator into the original limit:  
\[
\lim_{x \to 0} \frac{\sqrt{x+2} - \sqrt{2}}{x(1+x)} = \lim_{x \to 0} \frac{\frac{x}{\sqrt{x+2} + \sqrt{2}}}{x(1+x)}.
\]

Simplify the fraction by dividing \( x \) out:  
\[
\lim_{x \to 0} \frac{x}{\sqrt{x+2} + \sqrt{2}} \cdot \frac{1}{x(1+x)} = \lim_{x \to 0} \frac{1}{(\sqrt{x+2} + \sqrt{2})(1+x)}.
\]

---

### Step 3: Evaluate the limit as \( x \to 0 \)  
As \( x \to 0 \), the terms simplify as follows:  
- \( \sqrt{x+2} \to \sqrt{2} \),  
- \( 1 + x \to 1 \).

Thus, the denominator becomes:  
\[
(\sqrt{x+2} + \sqrt{2})(1+x) \to (\sqrt{2} + \sqrt{2})(1) = 2\sqrt{2}.
\]

---

### Final Answer:  
\[
\lim_{x \to 0} \frac{\sqrt{x+2} - \sqrt{2}}{x(1+x)} = \frac{1}{2\sqrt{2}}.
\]

---

### Do you want further clarifications or details on any step?  
Here are 5 related questions to explore:  
1. How can we generalize limits involving square roots and conjugates?  
2. What happens if we encounter \( x \to a \) instead of \( x \to 0 \)?  
3. How does multiplying by conjugates simplify expressions?  
4. Can L'Hôpital's Rule also solve this limit?  
5. How do we handle more complex roots in limits?  

**Tip:** When dealing with square roots in limits, multiplying by the conjugate is a key strategy to simplify the expression.

lim (x→0) [sqrt(x+2) - sqrt(2)] / [x(1+x)]

Learn how to solve the limit lim (x→0) [sqrt(x+2) - sqrt(2)] / [x(1+x)] step-by-step. This problem involves simplifying square root differences using conjugates and evaluating the limit. Suitable for students studying calculus and advanced algebra.

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