Limit Simplification 83B7Ff

1. **State the problem:** We need to find the limit $$\lim_{x \to 4} \frac{x-4}{(x-4)(\sqrt{x}+2)}$$. 2. **Identify the issue:** Direct substitution of $x=4$ gives $$\frac{4-4}{(4-4)(\sqrt{4}+2)} = \frac{0}{0}$$ which is an indeterminate form. 3. **Simplify the expression:** Since $x-4$ appears in both numerator and denominator, and $x \neq 4$ near the limit, we can cancel $x-4$: $$\frac{\cancel{x-4}}{\cancel{x-4}(\sqrt{x}+2)} = \frac{1}{\sqrt{x}+2}$$ 4. **Evaluate the limit of the simplified expression:** Substitute $x=4$: $$\frac{1}{\sqrt{4}+2} = \frac{1}{2+2} = \frac{1}{4}$$ 5. **Conclusion:** The limit is $$\boxed{\frac{1}{4}}$$.