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To find the limit:

\[
\lim_{x \to \infty} \left(1 + 2x\right)^{\frac{1}{2 \ln x}}
\]

we can proceed with the following steps:

1. **Rewrite the Expression in Exponential Form:**
   Rewrite the expression as:
   \[
   \lim_{x \to \infty} e^{\frac{1}{2 \ln x} \cdot \ln(1 + 2x)}
   \]
   So now we need to evaluate:
   \[
   \lim_{x \to \infty} \frac{\ln(1 + 2x)}{2 \ln x}
   \]

2. **Analyze the Fraction Inside the Exponent:**
   Consider the limit:
   \[
   \lim_{x \to \infty} \frac{\ln(1 + 2x)}{2 \ln x}
   \]
   As \( x \to \infty \), \( \ln(1 + 2x) \) behaves like \( \ln(2x) \), which can be rewritten as \( \ln 2 + \ln x \). Thus:
   \[
   \lim_{x \to \infty} \frac{\ln(1 + 2x)}{2 \ln x} = \lim_{x \to \infty} \frac{\ln(2) + \ln(x)}{2 \ln x}
   \]

3. **Separate the Terms in the Limit:**
   Split the limit as follows:
   \[
   = \lim_{x \to \infty} \frac{\ln 2}{2 \ln x} + \lim_{x \to \infty} \frac{\ln x}{2 \ln x}
   \]
   The first term, \( \frac{\ln 2}{2 \ln x} \), approaches 0 as \( x \to \infty \). The second term simplifies to:
   \[
   \lim_{x \to \infty} \frac{\ln x}{2 \ln x} = \frac{1}{2}
   \]

4. **Combine Results:**
   Therefore:
   \[
   \lim_{x \to \infty} \frac{\ln(1 + 2x)}{2 \ln x} = 0 + \frac{1}{2} = \frac{1}{2}
   \]

5. **Apply the Result:**
   Substitute this result back into the exponential expression:
   \[
   \lim_{x \to \infty} e^{\frac{1}{2 \ln x} \cdot \ln(1 + 2x)} = e^{\frac{1}{2}} = \sqrt{e}
   \]

### Final Answer:
\[
\lim_{x \to \infty} \left(1 + 2x\right)^{\frac{1}{2 \ln x}} = \sqrt{e}
\]

Would you like further details on any step? 

Here are some related questions to explore:

1. What happens if we change the base of the logarithm in this problem?
2. How would this problem change if the exponent were \( \frac{1}{3 \ln x} \) instead?
3. What methods can we use to find limits involving powers of functions?
4. How does L'Hôpital's Rule apply to limits with logarithmic expressions?
5. Why does the approximation \( \ln(1 + 2x) \approx \ln(2x) \) work for large \( x \)?

**Tip:** Converting expressions with exponents into exponential form \( e^{f(x)} \) can simplify complex limits, especially when dealing with logarithms.

Find the limit: \( \lim_{{x \to \infty}} (1 + 2x)^{{1/(2 \ln x)}} \)

Learn how to find the limit of \( (1 + 2x)^{1/(2 \ln x)} \) as x approaches infinity. This solution involves key calculus concepts like limits, exponential forms, and logarithmic approximations, making it ideal for undergraduate calculus students.

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