https://file.aiquickdraw.com/mathbot/file/1729016746559ajspg53w.jpg

The mathematical expression provided in the image appears to be:

\[
\lim_{x \to +\infty} \left( \frac{\sqrt{x+2} - 2}{\sqrt{x+7} - 3} \right)
\]

### Steps to Solve:

1. **Substitute \(x \to +\infty\):**
   As \(x\) approaches infinity, the dominant term in both the numerator and denominator is \( \sqrt{x} \). We can focus on simplifying the behavior as \(x\) becomes very large.

2. **Simplify using the limit properties:**
   To handle the infinity form, we multiply both the numerator and denominator by the conjugates of their respective terms to simplify the square roots:
   
   \[
   \frac{\sqrt{x+2} - 2}{\sqrt{x+7} - 3} \cdot \frac{\sqrt{x+2} + 2}{\sqrt{x+2} + 2} = \frac{(x+2) - 4}{(\sqrt{x+7} - 3)(\sqrt{x+7} + 3)}
   \]
   
   Simplify each part:
   \[
   = \frac{x - 2}{(x+7) - 9} = \frac{x - 2}{x - 2}
   \]
   
   For large \(x\), this simplifies to \(1\), because the lower-order terms (\(-2\), \(-9\)) become negligible as \(x \to \infty\).

### Conclusion:

\[
\lim_{x \to +\infty} \left( \frac{\sqrt{x+2} - 2}{\sqrt{x+7} - 3} \right) = 1
\]

---

Would you like a more detailed breakdown of any step, or do you have any specific questions?

Here are five related questions to expand your understanding:
1. What are common methods for solving limits involving infinity and square roots?
2. How do conjugates help simplify expressions involving square roots?
3. What is the impact of lower-order terms when evaluating limits at infinity?
4. How would this limit change if the coefficients of \(x\) were different inside the square roots?
5. How would you apply L'Hopital's Rule to limits of indeterminate forms like this one?

**Tip:** When evaluating limits at infinity, always check if multiplying by a conjugate or simplifying leading terms makes the limit clearer.

Evaluate the limit of the expression as x approaches infinity: (sqrt(x + 2) - 2) / (sqrt(x + 7) - 3).

Learn how to solve the limit of (sqrt(x + 2) - 2) / (sqrt(x + 7) - 3) as x approaches infinity. This solution covers key concepts in limits, rationalization using conjugates, and evaluating behavior at infinity. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation