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The expression we need to evaluate is:

\[
\lim_{{x \to \infty}} \left( x e^{\frac{1}{x}} - x \right)^x
\]

Let's go through the solution step-by-step.

### Step 1: Simplify the Expression Inside the Limit
Observe that:

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

Since \( e^{\frac{1}{x}} \approx 1 + \frac{1}{x} \) when \( x \to \infty \) (using the Taylor expansion for \( e^{\frac{1}{x}} \) around 0), we have:

\[
e^{\frac{1}{x}} - 1 \approx \frac{1}{x}
\]

Thus,

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

So, as \( x \to \infty \), the expression \( x e^{\frac{1}{x}} - x \) approaches 1.

### Step 2: Rewrite the Limit
We now have:

\[
\lim_{{x \to \infty}} \left( x e^{\frac{1}{x}} - x \right)^x \approx \lim_{{x \to \infty}} 1^x
\]

Since \( 1^x = 1 \) for any value of \( x \), the limit approaches:

\[
\lim_{{x \to \infty}} 1 = 1
\]

### Final Answer
The solution to the limit is:

\[
\boxed{1}
\]

Would you like more details or have any questions?

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Here are five related questions to expand on this concept:

1. How would the solution change if we replaced \( e^{\frac{1}{x}} \) with another exponential function?
2. What if the exponent \( x \) were replaced by \( x^2 \) or \( x \ln x \)?
3. Can you apply the same approach if the limit involves \( \ln(x) \) instead of an exponential?
4. How would the Taylor expansion differ if we had \( e^{\frac{2}{x}} \) instead of \( e^{\frac{1}{x}} \)?
5. What are the conditions for approximating \( e^{f(x)} \approx 1 + f(x) \) as \( x \to \infty \)?

**Tip:** Taylor series expansions are powerful tools for approximating expressions involving exponentials and logarithms around certain points.

Evaluate \( \lim_{{x \to \infty}} \left( x e^{\frac{1}{x}} - x \right)^x \).

Explore the step-by-step solution for evaluating \( \lim_{{x \to \infty}} \left( x e^{\frac{1}{x}} - x \right)^x \). This problem covers advanced concepts in limits, exponential functions, and Taylor series expansion, suitable for students in Grades 11-12.

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