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The problem asks for the limit as \( x \) approaches infinity of the function \( \sqrt[3]{x^3 + x^2 + 1 - x} \).

Let's analyze the expression inside the cube root:

\[
f(x) = \sqrt[3]{x^3 + x^2 + 1 - x}
\]

### Step 1: Simplify the expression inside the cube root
The dominant term inside the cube root as \( x \) approaches infinity is \( x^3 \), because it's the highest degree term. Therefore, we can factor \( x^3 \) out of the expression inside the cube root:

\[
f(x) = \sqrt[3]{x^3(1 + \frac{1}{x} + \frac{1}{x^2} - \frac{1}{x^2})}
\]

Simplifying the expression inside the cube root:

\[
f(x) = \sqrt[3]{x^3 \left(1 + \frac{x^2 + 1 - x}{x^3}\right)}
\]

As \( x \) approaches infinity, the fraction terms \( \frac{1}{x} \), \( \frac{1}{x^2} \), and \( \frac{1}{x^3} \) approach zero:

\[
f(x) = \sqrt[3]{x^3 \cdot 1} = \sqrt[3]{x^3} = x
\]

### Step 2: Evaluate the limit
As \( x \) approaches infinity:

\[
\lim_{x \to \infty} f(x) = \lim_{x \to \infty} x = \infty
\]

Therefore, the limit of the given function as \( x \) approaches infinity is \( \infty \).

Would you like a more detailed explanation, or do you have any questions? Here are some related questions you might find interesting:

1. How do you evaluate limits at infinity involving polynomial expressions?
2. What happens to functions involving roots as \( x \) approaches infinity?
3. How do you determine the dominant term in a polynomial?
4. Can limits at infinity ever result in a finite number?
5. How does L'Hôpital's Rule apply to limits involving infinity?

**Tip:** When finding limits involving infinity, focusing on the highest-degree term can often simplify the problem significantly.

Learn how to evaluate the limit as x approaches infinity of the function sqrt[3](x^3 + x^2 + 1 - x) with a detailed step-by-step solution. This problem covers limits involving polynomial expressions and the behavior of functions at infinity. Suitable for advanced high school students and calculus learners.

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