https://file.aiquickdraw.com/mathbot/file/1729872228070hz11z8fb.jpg

The given expression in the image is the limit:

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

Let's simplify this limit step by step.

1. **Rewrite the terms**: 
   \[
   \frac{\sqrt{x^2 + x + 1}}{x} = \frac{1}{x} \sqrt{x^2 + x + 1}
   \]
   \[
   \frac{x + 1}{x} = 1 + \frac{1}{x}
   \]

2. **Simplify the square root term**:
   Factor out \(x^2\) from inside the square root:
   \[
   \sqrt{x^2 + x + 1} = x\sqrt{1 + \frac{1}{x} + \frac{1}{x^2}}
   \]
   So the term becomes:
   \[
   \frac{\sqrt{x^2 + x + 1}}{x} = \sqrt{1 + \frac{1}{x} + \frac{1}{x^2}}
   \]

3. **Substitute into the limit**:
   Now the limit is:
   \[
   \lim_{x \to \infty} \left( \sqrt{1 + \frac{1}{x} + \frac{1}{x^2}} - \left(1 + \frac{1}{x}\right) \right)
   \]

4. **Evaluate the limit as \(x \to \infty\)**:
   As \(x\) approaches infinity, the terms \(\frac{1}{x}\) and \(\frac{1}{x^2}\) approach 0. Thus, the expression simplifies to:
   \[
   \lim_{x \to \infty} \left( \sqrt{1 + 0 + 0} - \left(1 + 0\right) \right) = \sqrt{1} - 1 = 0
   \]

Therefore, the value of the limit is:

\[
\boxed{0}
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. What happens if the higher-order terms are not considered negligible in limits?
2. How would the solution change if the numerator had a different form (e.g., \(x^2 + 2x + 1\))?
3. Why is factoring \(x^2\) from the square root important in this process?
4. How does this problem relate to limits involving infinity in other calculus problems?
5. What role do higher-order terms play when approaching limits?

**Tip**: When dealing with limits at infinity, always check if you can factor out dominant terms to simplify the expression.

Evaluate the limit as x approaches infinity of the expression: (sqrt(x^2 + x + 1)/x - (x + 1)/x)

This problem involves evaluating the limit as x approaches infinity of the expression (sqrt(x^2 + x + 1)/x - (x + 1)/x). The solution uses simplification techniques and dominant term analysis to show that the limit equals 0. Suitable for students studying Calculus.

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