# Why does l'hopitals rule here give me the wrong answer?

$$\lim_{x \to 4}\frac{\sqrt{5-x} - 1}{2-\sqrt{x}}.$$

Using l'hopitals rule here, I end up getting:

top derivative: $-1 / 2(5-x)^{1/2}$

bottom: $2(5-x)^{1/2}$

Thus getting the answer of $-2$, when putting 4 in the entire thing. Am I applying the method incorrectly?

The answer is $2$.

It looks like you're taking the derivative of the denominator incorrectly: $$(2-\sqrt{x})'=(2-x^{1\over 2})'=-{1\over 2}x^{-{1\over 2}}.$$ So the whole thing simplifies to $$\lim_{x\rightarrow 4}{x^{1\over 2}\over (5-x)^{1\over 2}},$$ which gives $2$ as expected.