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?

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.