https://file.aiquickdraw.com/mathbot/file/17256026098883b4l9hx6.jpg

The expression from the image was recognized as:

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

Let's solve this limit step by step.

### Step-by-step solution:

1. **Initial expression**: 
   \[
   \lim_{{x \to \infty}} \left( \sqrt{x^2 + 1} - x \right)
   \]
   Direct substitution of \( x \to \infty \) results in an indeterminate form \( \infty - \infty \). So we need to manipulate the expression.

2. **Rationalizing the expression**: 
   Multiply and divide by the conjugate to simplify:
   \[
   \lim_{{x \to \infty}} \left( \sqrt{x^2 + 1} - x \right) \cdot \frac{\sqrt{x^2 + 1} + x}{\sqrt{x^2 + 1} + x}
   \]
   This becomes:
   \[
   \lim_{{x \to \infty}} \frac{\left( \sqrt{x^2 + 1} \right)^2 - x^2}{\sqrt{x^2 + 1} + x}
   \]
   Simplifying the numerator:
   \[
   \left( \sqrt{x^2 + 1} \right)^2 - x^2 = x^2 + 1 - x^2 = 1
   \]
   So the expression reduces to:
   \[
   \lim_{{x \to \infty}} \frac{1}{\sqrt{x^2 + 1} + x}
   \]

3. **Evaluate the limit**: 
   As \( x \to \infty \), \( \sqrt{x^2 + 1} \) behaves like \( x \), so:
   \[
   \lim_{{x \to \infty}} \frac{1}{\sqrt{x^2 + 1} + x} \approx \lim_{{x \to \infty}} \frac{1}{2x}
   \]
   As \( x \to \infty \), \( \frac{1}{2x} \to 0 \).

Therefore, the limit is:

\[
\boxed{0}
\]

Would you like further details or have any questions about this solution? 

Here are 5 related questions for further exploration:
1. What happens if the term inside the square root is \( x^2 + c \), where \( c \) is a constant?
2. How would the result change if the limit was as \( x \to -\infty \)?
3. Can you apply L'Hôpital's Rule to this problem, and how would it affect the process?
4. How can this limit be interpreted geometrically on a graph?
5. What are some common techniques to solve limits that yield indeterminate forms like \( \infty - \infty \)?

**Tip**: Always check for indeterminate forms when solving limits, and use techniques like rationalization or factoring to simplify them!

Find the limit as x approaches infinity for the expression sqrt(x^2 + 1) - x.

Explore the limit of the expression sqrt(x^2 + 1) - x as x approaches infinity, including a step-by-step solution using rationalization to simplify the limit expression to zero. This advanced topic is suitable for university-level students studying calculus.

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