<s>[INST] $\lim _{x\to 4}\frac{\sqrt{5-x}-1}{2-\sqrt{x}}$ Find the limit of $f(x)=\frac{\sqrt{5-x}-1}{2-\sqrt{x}}$ when x approaches 4.
I have no idea where to start with this problem.
 [/INST] Without using L'Hospital's rule
Notice, 
Rationalizing the numerator & denominator as follows
$$\lim_{x\to 4}\frac{\sqrt{5-x}-1}{2-\sqrt{x}}$$ 
$$=\lim_{x\to 4}\frac{(\sqrt{5-x}-1)(\sqrt{5-x}+1)(2+\sqrt{x})}{(2-\sqrt{x})(2+\sqrt{x})(\sqrt{5-x}+1)}$$ 
$$=\lim_{x\to 4}\frac{(5-x-1)(2+\sqrt{x})}{(4-x)(\sqrt{5-x}+1)}$$ 
$$=\lim_{x\to 4}\frac{(4-x)(2+\sqrt{x})}{(4-x)(\sqrt{5-x}+1)}$$ 
$$=\lim_{x\to 4}\frac{2+\sqrt{x}}{\sqrt{5-x}+1}$$ 
$$=\frac{2+\sqrt{4}}{\sqrt{5-4}+1}=\frac{4}{2}=\color{blue}{2}$$ 
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