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We need to evaluate the limit:

\[
\lim_{x \to \infty} \left( e^x + x \right)^{1/x}.
\]

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### Solution:

1. **Factor out \( e^x \) from the term \( e^x + x \):**

\[
e^x + x = e^x \left( 1 + \frac{x}{e^x} \right).
\]

This helps us split the expression into two parts.

2. **Rewrite the given limit:**

\[
\left( e^x + x \right)^{1/x} = \left[ e^x \left( 1 + \frac{x}{e^x} \right) \right]^{1/x}.
\]

Using the property of exponents \( (ab)^n = a^n \cdot b^n \), we get:

\[
\left( e^x + x \right)^{1/x} = \left( e^x \right)^{1/x} \cdot \left( 1 + \frac{x}{e^x} \right)^{1/x}.
\]

3. **Simplify each term:**

- First term: \( \left( e^x \right)^{1/x} \):

\[
\left( e^x \right)^{1/x} = e^{x \cdot \frac{1}{x}} = e^1 = e.
\]

- Second term: \( \left( 1 + \frac{x}{e^x} \right)^{1/x} \):

As \( x \to \infty \), note that \( \frac{x}{e^x} \to 0 \) because \( e^x \) grows much faster than \( x \). Therefore:

\[
1 + \frac{x}{e^x} \to 1.
\]

Taking the \( 1/x \)-th power:

\[
\left( 1 + \frac{x}{e^x} \right)^{1/x} \to 1.
\]

4. **Combine the results:**

Now combining both terms, we get:

\[
\left( e^x + x \right)^{1/x} = e \cdot 1 = e.
\]

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### Final Answer:

\[
\lim_{x \to \infty} \left( e^x + x \right)^{1/x} = e.
\]

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Would you like a deeper explanation of any step? Let me know!

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### Related Questions:
1. What happens if \( x \to \infty \) for a similar expression involving \( e^{2x} \)?
2. How does the rate of growth compare between \( e^x \) and polynomial terms like \( x^n \)?
3. How do we compute limits involving \( \log(x) \) or other logarithmic terms?
4. What is the general method for solving exponential and logarithmic limits?
5. How does the dominance of \( e^x \) help simplify complicated limits?

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### Tip:
When working with exponential and polynomial terms, always factor out the dominant term to simplify the expression.

Learn how to evaluate the limit lim (x→∞) (e^x + x)^(1/x) using dominant term analysis and exponential growth concepts. This step-by-step solution explains why e dominates polynomials as x approaches infinity.

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