The expression you've provided is:

$\lim_{{x \to \infty}} \frac{\sqrt{x + 1} - \sqrt{x}}{1}$

and it seems to imply that the limit is equal to $\frac{1}{2}$.

Let's solve this step-by-step.

Step-by-Step Solution

1. Original Expression: $\lim_{{x \to \infty}} \frac{\sqrt{x + 1} - \sqrt{x}}{1}$

2. Multiply by the Conjugate: To simplify, we multiply the numerator and the denominator by the conjugate of the numerator, which is $\sqrt{x + 1} + \sqrt{x}$: $\lim_{{x \to \infty}} \frac{\left(\sqrt{x + 1} - \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}$

3. Simplify the Numerator: Using the difference of squares, $(a - b)(a + b) = a^2 - b^2$, we get: $= \lim_{{x \to \infty}} \frac{(x + 1) - x}{\sqrt{x + 1} + \sqrt{x}}$

   Simplifying further, the $x$ terms cancel: $= \lim_{{x \to \infty}} \frac{1}{\sqrt{x + 1} + \sqrt{x}}$

4. Evaluate the Denominator: As $x$ approaches infinity, both $\sqrt{x + 1}$ and $\sqrt{x}$ behave like $\sqrt{x}$. So: $\sqrt{x + 1} + \sqrt{x} \approx 2\sqrt{x}$

   Thus, the expression becomes: $\lim_{{x \to \infty}} \frac{1}{2\sqrt{x}}$

5. Take the Limit: As $x \to \infty$, $\frac{1}{2\sqrt{x}} \to 0$.

Conclusion

The limit is: $\lim_{{x \to \infty}} \frac{\sqrt{x + 1} - \sqrt{x}}{1} = 0$

Therefore, the correct answer is $0$, not $\frac{1}{2}$.

Would you like more details or have any questions?

Here are five related questions to deepen understanding:

1. Why do we use the conjugate method when simplifying expressions with square roots in limits?
2. How does the behavior of $\sqrt{x}$ as $x$ approaches infinity impact limits?
3. What other techniques, besides conjugate multiplication, can be used for evaluating limits involving radicals?
4. How would the limit change if it was $\sqrt{x + c} - \sqrt{x}$ for some constant $c$?
5. What is the significance of higher-order terms when evaluating limits of functions at infinity?

Tip: When dealing with limits at infinity, always consider simplifying expressions using techniques like conjugate multiplication or factoring dominant terms.