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To solve the limit:

\[
\lim_{{x \to \infty}} \sqrt{x^2 + x + 1} + x
\]

we need to analyze the behavior of the expression as \( x \) approaches infinity.

1. **Rewrite the expression under the square root:**
   Observe that \( x^2 \) is the dominant term inside the square root when \( x \to \infty \). To make it clearer, we can factor \( x^2 \) out of the terms inside the square root:

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

2. **Take the limit inside the square root:**
   As \( x \to \infty \), the terms \( \frac{1}{x} \) and \( \frac{1}{x^2} \) approach zero. So,

   \[
   \sqrt{1 + \frac{1}{x} + \frac{1}{x^2}} \to \sqrt{1} = 1
   \]

   This means that as \( x \to \infty \),

   \[
   \sqrt{x^2 + x + 1} \approx x
   \]

3. **Rewrite the whole expression with this approximation:**
   Substitute back into the limit expression:

   \[
   \lim_{{x \to \infty}} \left(\sqrt{x^2 + x + 1} + x\right) \approx \lim_{{x \to \infty}} \left(x + x\right) = \lim_{{x \to \infty}} 2x
   \]

4. **Evaluate the limit:**
   As \( x \to \infty \), \( 2x \to \infty \).

Thus,

\[
\lim_{{x \to \infty}} \sqrt{x^2 + x + 1} + x = \infty
\]

The limit diverges to infinity.

Would you like further details on any part of this solution?

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Here are five related questions to deepen understanding:

1. What happens if we change \( x + 1 \) inside the square root to \( x^3 \)? How would that affect the limit?
2. How would the answer change if the term outside the square root was \( -x \) instead of \( +x \)?
3. Why is \( x^2 \) considered the dominant term inside the square root for large \( x \)?
4. How would this limit change if the expression was divided by \( x \)?
5. What techniques are generally useful for evaluating limits involving square roots and infinity?

**Tip:** When analyzing limits involving square roots, factor out the highest power inside the square root to simplify the expression.

Evaluate the limit as x approaches infinity: lim (x -> ∞) √(x^2 + x + 1) + x

Learn to evaluate the limit lim (x -> ∞) √(x^2 + x + 1) + x by analyzing the dominant terms as x approaches infinity. This solution demonstrates factoring and asymptotic analysis techniques, suitable for college-level calculus students.

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