https://file.aiquickdraw.com/mathbot/file/1729701660469o4myalqh.jpg

The problem in the image is the following limit expression:

\[
\lim_{{x \to +\infty}} \frac{x - \sqrt{x^2 + x + 1}}{2x + \sqrt{4x^2 + x}}
\]

To solve this limit as \( x \to +\infty \), we use asymptotic analysis and simplifying techniques. The key idea is to factor out terms that dominate as \( x \to \infty \) and deal with the square roots.

### Step-by-step solution:

#### 1. Focus on the numerator:
\[
x - \sqrt{x^2 + x + 1}
\]

Factor out \( x \) from the square root:
\[
\sqrt{x^2 + x + 1} = x \sqrt{1 + \frac{1}{x} + \frac{1}{x^2}}
\]
As \( x \to \infty \), the terms \( \frac{1}{x} \) and \( \frac{1}{x^2} \) become negligible, so:
\[
\sqrt{x^2 + x + 1} \sim x \left( 1 + \frac{1}{2x} \right)
\]
Thus, the expression becomes:
\[
x - x \left( 1 + \frac{1}{2x} \right) = x - x - \frac{1}{2} = -\frac{1}{2}
\]
So, the numerator approaches \(-\frac{1}{2}\) as \( x \to \infty \).

#### 2. Focus on the denominator:
\[
2x + \sqrt{4x^2 + x}
\]
Again, factor out \( x \) from the square root:
\[
\sqrt{4x^2 + x} = 2x \sqrt{1 + \frac{1}{4x}}
\]
As \( x \to \infty \), \( \frac{1}{4x} \) becomes negligible, so:
\[
\sqrt{4x^2 + x} \sim 2x \left( 1 + \frac{1}{8x} \right)
\]
Thus, the denominator becomes:
\[
2x + 2x \left( 1 + \frac{1}{8x} \right) = 2x + 2x + \frac{1}{4} = 4x + \frac{1}{4}
\]
So, the denominator approaches \( 4x \) as \( x \to \infty \).

#### 3. Combine the results:
Now, the limit can be simplified:
\[
\lim_{{x \to +\infty}} \frac{-\frac{1}{2}}{4x} = 0
\]

### Final answer:
\[
\lim_{{x \to +\infty}} \frac{x - \sqrt{x^2 + x + 1}}{2x + \sqrt{4x^2 + x}} = 0
\]

Would you like more details on any part of the process?

### Related questions:
1. How does the technique of factoring out dominant terms help in limits involving square roots?
2. What are the most common ways to simplify expressions involving large \(x\)?
3. Why do terms like \( \frac{1}{x} \) and \( \frac{1}{x^2} \) become negligible as \( x \to \infty \)?
4. Can this limit be solved using L'Hôpital's Rule instead? How?
5. What are the potential pitfalls of not simplifying correctly in limits with square roots?

### Tip:
When dealing with limits as \( x \to \infty \), always consider the highest-order terms first, as lower-order terms often become negligible.

Find the limit: lim(x -> +∞) of the expression (x - sqrt(x^2 + x + 1)) / (2x + sqrt(4x^2 + x))

This problem explores the limit of the expression (x - sqrt(x^2 + x + 1)) / (2x + sqrt(4x^2 + x)) as x approaches infinity. The solution involves asymptotic analysis, factoring dominant terms, and simplifying square roots. Suitable for undergraduate and graduate students studying calculus.

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