$$ \lim_{x \to \infty} x^{1/x} = e^{\lim_{x \to \infty} \frac{\ln x}{x}} = e^0 = 1 $$

The given expression is the limit of $ x^{1/x} $ as $ x \to \infty $. Let me walk through the steps of solving this limit:

### Step-by-Step Solution:

1. **Expression Setup**:
   The original limit is $ \lim_{x \to \infty} x^{1/x} $. To solve this, we take the natural logarithm to simplify the exponent:
   \[
   y = x^{1/x}
   \]
   Taking the natural log of both sides:
   \[
   \ln y = \ln \left( x^{1/x} \right) = \frac{1}{x} \ln x
   \]
   Now, we need to find the limit of $ \ln y $ as $ x \to \infty $.

2. **Limit of the Logarithmic Expression**:
   \[
   \lim_{x \to \infty} \ln y = \lim_{x \to \infty} \frac{\ln x}{x}
   \]
   This is a well-known limit that can be evaluated using L'Hopital's Rule, since it is an indeterminate form $ \frac{\infty}{\infty} $.

3. **Applying L'Hopital's Rule**:
   Differentiate the numerator and the denominator:
   \[
   \text{Numerator: } \frac{d}{dx} \ln x = \frac{1}{x}
   \]
   \[
   \text{Denominator: } \frac{d}{dx} x = 1
   \]
   Now, applying L'Hopital's Rule:
   \[
   \lim_{x \to \infty} \frac{\ln x}{x} = \lim_{x \to \infty} \frac{1/x}{1} = \lim_{x \to \infty} \frac{1}{x} = 0
   \]

4. **Conclusion**:
   Since $ \lim_{x \to \infty} \ln y = 0 $, this means:
   \[
   \ln y = 0 \implies y = e^0 = 1
   \]
   Therefore, the original limit is:
   \[
   \lim_{x \to \infty} x^{1/x} = 1
   \]

### Final Answer:
\[
\lim_{x \to \infty} x^{1/x} = 1
\]

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What is L'Hopital's Rule and when can it be applied?
2. How do you find limits involving logarithmic and exponential functions?
3. What are some other common limits that tend to 1 as $ x \to \infty $?
4. How does logarithmic differentiation help simplify certain limit problems?
5. What is the significance of the base $ e $ in limit problems involving logarithms?

**Tip**: Whenever you encounter a function raised to a power involving $ x $, logarithmic differentiation can often simplify the expression before applying limits.

Discover how to find the limit of x^(1/x) as x approaches infinity with this detailed step-by-step solution. Explore key concepts such as L'Hopital's Rule, logarithmic differentiation, and limits involving exponential and logarithmic functions. Suitable for students in Grades 11-12.

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