https://file.aiquickdraw.com/mathbot/file/17263037967663380r6g2.jpg

It looks like you've provided an image with some handwritten mathematical work. The main part of the image seems to be working through a limit problem involving a square root function.

Let's break down what I can extract from the image:

### Problem:
You appear to be solving:

\[
\lim_{x \to 4} \frac{3 - \sqrt{x + 5}}{x - 4}
\]

### Steps Extracted:
1. **Step 1**: The expression starts as:

\[
\frac{3 - \sqrt{x + 5}}{x - 4}
\]

2. **Step 2**: The next step involves multiplying both the numerator and denominator by the conjugate of the numerator:

\[
\frac{3 - \sqrt{x + 5}}{x - 4} \times \frac{3 + \sqrt{x + 5}}{3 + \sqrt{x + 5}}
\]

3. **Step 3**: After multiplying out, you obtain a difference of squares in the numerator:

\[
\frac{9 - (x + 5)}{(x - 4)(3 + \sqrt{x + 5})}
\]

Simplifying the numerator gives:

\[
\frac{4 - x}{(x - 4)(3 + \sqrt{x + 5})}
\]

4. **Step 4**: Notice that \(4 - x = -(x - 4)\), so you can cancel out the \(x - 4\) terms, leading to:

\[
\frac{-1}{3 + \sqrt{x + 5}}
\]

5. **Step 5**: Now, substitute \(x = 4\) directly into the simplified expression:

\[
\frac{-1}{3 + \sqrt{4 + 5}} = \frac{-1}{3 + 3} = \frac{-1}{6}
\]

### Final Answer:
The limit is:

\[
\lim_{x \to 4} \frac{3 - \sqrt{x + 5}}{x - 4} = \frac{-1}{6}
\]

---

Let me know if you'd like a more detailed explanation or have any questions about the steps.

Here are some related questions you might explore:
1. How do conjugates help simplify expressions involving square roots?
2. Why is it important to cancel terms carefully when simplifying limits?
3. What is the general strategy for solving limits involving square roots?
4. Could this problem be solved using L'Hopital's Rule instead?
5. How does factoring help when simplifying expressions before applying limits?

**Tip**: Always look for opportunities to use the conjugate method when you have square roots in limit problems, as it helps eliminate irrational terms.

Explore how to solve the limit lim x->4 (3 - √(x+5))/(x-4) using conjugate multiplication and simplification techniques. This detailed solution covers important mathematical concepts such as limits and rationalizing square roots, suitable for students in Grades 10-12.

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