https://file.aiquickdraw.com/mathbot/file/1727001618716vb2p5ro0.jpg

The expression in the image is:

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

Let’s solve this step by step.

### Step 1: Simplify the square root terms as \(x \to +\infty\)

For large \(x\), we approximate the square roots:
1. \(\sqrt{x^2 + 2x} = \sqrt{x^2(1 + \frac{2}{x})} = x\sqrt{1 + \frac{2}{x}} \approx x\left(1 + \frac{1}{x}\right) = x + 1\), since for large \(x\), \(\sqrt{1 + \frac{2}{x}} \approx 1 + \frac{1}{x}\).
2. \(\sqrt{x^2 + x + x} = \sqrt{x^2 + 2x} \approx x + 1\), using the same approximation as in (1).

### Step 2: Substitute the approximations

Now substitute these approximations into the limit expression:
\[
\lim_{x \to +\infty} x \left( (x + 1) - 2(x + 1) \right)
\]

Simplifying inside the parentheses:
\[
(x + 1) - 2(x + 1) = x + 1 - 2x - 2 = -x - 1
\]

Thus, the expression becomes:
\[
\lim_{x \to +\infty} x(-x - 1) = \lim_{x \to +\infty} (-x^2 - x)
\]

### Step 3: Evaluate the limit

As \(x \to +\infty\), \(-x^2 - x\) clearly tends to \(-\infty\).

Thus, the final result is:

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

Would you like further details or clarifications?

### Related Questions:
1. How do we handle limits that approach infinity with multiple terms?
2. Why is the approximation of square roots valid for large values of \(x\)?
3. How would the answer change if the coefficients inside the square roots were different?
4. What if the function inside the square roots had higher powers of \(x\)?
5. How does this technique apply to functions involving cube roots or other powers?

### Tip:
When evaluating limits at infinity, approximate the dominant terms (highest powers) to simplify the expression and focus on leading behaviors.

What is the limit of x(√(x² + 2x) - 2√(x² + x + x)) as x approaches infinity?

Learn how to evaluate the limit of x(√(x² + 2x) - 2√(x² + x + x)) as x approaches infinity. This problem covers limits, asymptotic approximations, and algebraic simplifications for higher-level students. Suitable for Grades 10-12.

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