We are tasked with finding the limit:

$\lim_{x \to \infty} \left( \sqrt{x^2 + 20x} - \sqrt{x^2 - 20x + 2024} \right)$

Solution:

1. Factor out $x^2$ from both square roots to simplify the expression for large $x$. We know that for large $x$, the dominant term in each square root will be $x^2$, so we can factor it out:

$\sqrt{x^2 + 20x} = x \sqrt{1 + \frac{20}{x}} \quad \text{and} \quad \sqrt{x^2 - 20x + 2024} = x \sqrt{1 - \frac{20}{x} + \frac{2024}{x^2}}$

2. Apply the limit as $x \to \infty$. Both expressions inside the square roots tend to 1 as $x \to \infty$, so we can perform an expansion using the binomial approximation for small terms:

   • $\sqrt{1 + \frac{20}{x}} \approx 1 + \frac{10}{x}$
   • $\sqrt{1 - \frac{20}{x} + \frac{2024}{x^2}} \approx 1 - \frac{10}{x} + \frac{1012}{x^2}$

3. Subtract the two terms:

$x \left( \sqrt{1 + \frac{20}{x}} - \sqrt{1 - \frac{20}{x} + \frac{2024}{x^2}} \right) \approx x \left( \left( 1 + \frac{10}{x} \right) - \left( 1 - \frac{10}{x} + \frac{1012}{x^2} \right) \right)$

Simplifying the subtraction:

$x \left( \frac{10}{x} + \frac{10}{x} - \frac{1012}{x^2} \right) = 20 - \frac{1012}{x}$

4. Take the limit as $x \to \infty$:

$\lim_{x \to \infty} \left( 20 - \frac{1012}{x} \right) = 20$

Final Answer:

$\lim_{x \to \infty} \left( \sqrt{x^2 + 20x} - \sqrt{x^2 - 20x + 2024} \right) = 20$

Would you like any further details or have any questions?

Here are five questions related to the topic:

1. How do binomial approximations help simplify limits involving square roots?
2. Can you explain how limits at infinity affect dominant terms in expressions?
3. What other methods can be used to simplify square roots when dealing with limits?
4. Why does the $\frac{1012}{x^2}$ term become negligible as $x$ approaches infinity?
5. How would the result differ if the constant in the second square root was larger?

Tip: When simplifying square root expressions, factoring out the highest power of $x$ often makes it easier to find the limit at infinity.