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We are tasked with evaluating the limit:

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

To simplify this expression and find the limit, we can multiply by the conjugate of the expression. This will help eliminate the square roots:

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

Multiplying both the numerator and denominator by the conjugate, we get:

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

Simplifying the terms in the numerator:

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

Now let's analyze this expression as \( x \to \infty \):

- The numerator approaches \( -x \), since \( -x + 1 \approx -x \) for large \( x \).
- The denominator approaches \( \sqrt{2x} \), because \( \sqrt{x + 1} \approx \sqrt{x} \) and \( \sqrt{x + 1} + \sqrt{2x} \approx \sqrt{2x} + \sqrt{x} \).

Thus, the expression becomes:

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

Factor \( \sqrt{x} \) out of the denominator:

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

As \( x \to \infty \), \( \sqrt{x} \to \infty \), so the entire expression tends to \( -\infty \). Therefore, the limit is:

\[
\boxed{-\infty}
\]

Let me know if you want further details or have any questions.

Here are 5 questions to deepen your understanding:
1. Why do we use the conjugate method when dealing with square roots in limits?
2. How does factoring out \( \sqrt{x} \) simplify the expression?
3. What happens to the terms inside the square root as \( x \to \infty \)?
4. What are some other methods to evaluate limits with square roots?
5. Can the limit of a square root expression approach a finite number when \( x \to \infty \)?

**Tip:** Multiplying by the conjugate is a powerful tool for simplifying expressions involving square roots, especially when approaching limits at infinity.

Évalue cette limite: lim (x->∞) (sqrt(x+1) - sqrt(2x))

Explore the step-by-step process of evaluating the limit of sqrt(x+1) - sqrt(2x) as x approaches infinity. This article breaks down the use of the conjugate method to simplify square roots, providing a clear pathway to understanding how limits operate at infinity.

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