https://file.aiquickdraw.com/mathbot/file/17266911318806aq8ogh9.jpg

The mathematical expression in the image is:

\[
\lim_{{x \to 2}} \frac{\sqrt{1 + 2x + 3} - 2}{x - 2}
\]

This is a limit problem that can be solved using the technique of rationalizing the numerator. Let's go through the steps to solve it.

### Step-by-Step Solution:

1. **Rationalize the numerator**: We multiply both the numerator and denominator by the conjugate of the numerator.

   The conjugate of \( \sqrt{1 + 2x + 3} - 2 \) is \( \sqrt{1 + 2x + 3} + 2 \). So, multiply both the numerator and denominator by this conjugate:
   
   \[
   \lim_{{x \to 2}} \frac{(\sqrt{1 + 2x + 3} - 2)(\sqrt{1 + 2x + 3} + 2)}{(x - 2)(\sqrt{1 + 2x + 3} + 2)}
   \]

2. **Simplify the numerator**: The numerator becomes a difference of squares:
   
   \[
   (\sqrt{1 + 2x + 3})^2 - 2^2 = (1 + 2x + 3) - 4 = 2x
   \]
   
   So now the expression is:

   \[
   \lim_{{x \to 2}} \frac{2x}{(x - 2)(\sqrt{1 + 2x + 3} + 2)}
   \]

3. **Simplify the expression**: Factor out \(x - 2\) from the numerator.

   Since \(2x = 2(x - 2 + 2)\), we have \(2x = 2((x - 2) + 4)\), allowing us to cancel out the \(x - 2\) terms:

   After cancellation, we get:

   \[
   \lim_{{x \to 2}} \frac{2}{\sqrt{1 + 2x + 3} + 2}
   \]

4. **Substitute \(x = 2\)**: Finally, substitute \(x = 2\) into the expression:

   \[
   \frac{2}{\sqrt{1 + 2(2) + 3} + 2} = \frac{2}{\sqrt{1 + 4 + 3} + 2} = \frac{2}{\sqrt{8} + 2} = \frac{2}{2\sqrt{2} + 2}
   \]

5. **Simplify the final expression**:

   Factor out the 2 in the denominator:
   
   \[
   \frac{2}{2(\sqrt{2} + 1)} = \frac{1}{\sqrt{2} + 1}
   \]
   
   To further simplify, multiply both numerator and denominator by \(\sqrt{2} - 1\) to rationalize the denominator:

   \[
   \frac{1}{\sqrt{2} + 1} \cdot \frac{\sqrt{2} - 1}{\sqrt{2} - 1} = \frac{\sqrt{2} - 1}{(\sqrt{2})^2 - 1^2} = \frac{\sqrt{2} - 1}{2 - 1} = \sqrt{2} - 1
   \]

Thus, the value of the limit is:

\[
\boxed{\sqrt{2} - 1}
\]

Let me know if you need further details on this solution!

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Here are five related questions to deepen your understanding:
1. What happens if you try to solve this limit without rationalizing the numerator?
2. How would you apply L'Hopital's Rule to this problem instead of rationalizing the numerator?
3. Can you find the derivative of the function \( f(x) = \sqrt{1 + 2x + 3} \) using first principles?
4. What is the general approach to solving limits involving square roots?
5. How do you rationalize the denominator if it contains a square root?

**Tip**: Rationalizing is a common technique in limits involving square roots. Always look for the conjugate to simplify the expression effectively!

Find the limit: lim (x → 2) [ (sqrt(1 + 2x + 3) - 2) / (x - 2) ]

Solve the limit lim (x → 2) [ (sqrt(1 + 2x + 3) - 2) / (x - 2) ] by rationalizing the numerator. Learn step-by-step techniques involving square roots, conjugates, and simplification methods. Suitable for pre-calculus students.

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