The image shows the solution to a limit problem, which appears to involve a square root in the numerator. Let's go through the steps to clarify the solution:

Problem:

$\lim_{x \to 9} \frac{3 - \sqrt{x+5}}{x - 4}$

Solution Steps:

1. Rationalize the Numerator: The first step is to multiply the numerator and the denominator by the conjugate of the numerator to simplify the expression. This conjugate is $3 + \sqrt{x+5}$.

   $\lim_{x \to 9} \frac{(3 - \sqrt{x+5})(3 + \sqrt{x+5})}{(x-4)(3 + \sqrt{x+5})}$

2. Expand and Simplify: Multiply out the numerator using the difference of squares formula: $(3 - \sqrt{x+5})(3 + \sqrt{x+5}) = 3^2 - (\sqrt{x+5})^2 = 9 - (x + 5)$

   $= 9 - x - 5 = 4 - x$

   So now the expression becomes: $\lim_{x \to 9} \frac{4 - x}{(x-4)(3 + \sqrt{x+5})}$

3. Further Simplification: Since $4 - x$ is $-(x - 4)$: $\lim_{x \to 9} \frac{-(x - 4)}{(x-4)(3 + \sqrt{x+5})}$

   Cancel out $(x - 4)$: $\lim_{x \to 9} \frac{-1}{3 + \sqrt{x+5}}$

4. Evaluate the Limit: Substitute $x = 9$ into the simplified expression: $= \frac{-1}{3 + \sqrt{9+5}} = \frac{-1}{3 + \sqrt{14}}$

So, the final result is: $\lim_{x \to 9} \frac{3 - \sqrt{x+5}}{x - 4} = \frac{-1}{3 + \sqrt{14}}$

Would you like more details or have any questions?

Additional Questions:

1. What is the conjugate and why is it used in this problem?
2. How does the difference of squares help in simplifying the expression?
3. Why was it necessary to cancel out the $(x-4)$ term?
4. How do you know when to rationalize the numerator versus the denominator?
5. Could you solve this limit using L'Hôpital's Rule, and how would it compare to this method?

Tip:

When facing a limit problem with a square root, rationalizing by multiplying by the conjugate is often a helpful strategy to simplify the expression and avoid indeterminate forms.