https://file.aiquickdraw.com/mathbot/file/1726260797580m594prjm.jpg

The mathematical expression in the image is:

\[
\lim_{x \to 1} \frac{(2 + \sqrt{x})\sqrt{2 - x} - 3}{x^2 - 1}
\]

To solve this limit, let's first substitute \(x = 1\) directly into the expression and check for any indeterminate form:

1. The numerator becomes:
   \[
   (2 + \sqrt{1})\sqrt{2 - 1} - 3 = (2 + 1)(1) - 3 = 3 - 3 = 0
   \]
2. The denominator becomes:
   \[
   1^2 - 1 = 1 - 1 = 0
   \]

Since both the numerator and denominator result in 0, we have a **0/0** indeterminate form. Therefore, we need to apply **L'Hôpital's Rule**, which requires differentiating the numerator and denominator separately with respect to \(x\).

Let me proceed with that calculation.After applying L'Hôpital's Rule, the limit of the given expression as \( x \to 1 \) is:

\[
\lim_{x \to 1} \frac{(2 + \sqrt{x})\sqrt{2 - x} - 3}{x^2 - 1} = -\frac{1}{2}
\]

Would you like further details on how the derivatives were computed, or do you have any other questions?

Here are 5 related questions you might find interesting:
1. What is L'Hôpital's Rule, and when can it be applied?
2. Can you solve this limit using a series expansion approach?
3. How would this limit change if the powers of \(x\) in the denominator were different?
4. What are other techniques for solving limits involving square roots?
5. How can this problem be solved graphically or using a numerical approach?

**Tip**: L'Hôpital's Rule is useful for indeterminate forms like 0/0 or ∞/∞, but always check the conditions for applying it!

Find the limit as x approaches 1 for the expression: (2 + √x)√(2 - x) - 3 divided by x^2 - 1.

This problem explores the limit of (2 + √x)√(2 - x) - 3 as x approaches 1 using L'Hôpital's Rule. The solution involves differentiating both the numerator and denominator, ultimately resulting in a limit of -1/2. Ideal for university-level calculus students.

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