<s>[INST] How to evaluate the limit $\lim_{x\to0}\frac{\sqrt{x+1}-\sqrt{2x+1}}{\sqrt{3x+4}-\sqrt{2x+4}}$ First I tried direct substitution, which resulted in the indeterminate form. Then because of the square roots, I tried rationalizing (both numerator and denominator), but still get the indeterminate form. I can't use L'Hopital's rule because we haven't learned that yet (not sure if it would work anyway). Don't really know what else to try algebraically... Any hints?
 [/INST] $\lim_{x\to0}\frac{\sqrt{x+1}-\sqrt{2x+1}}{\sqrt{3x+4}-\sqrt{2x+4}}\cdot\frac{\sqrt{3x+4}+\sqrt{2x+4}}{\sqrt{3x+4}+\sqrt{2x+4}}\cdot\frac{\sqrt{x+1}+\sqrt{2x+1}}{\sqrt{x+1}+\sqrt{2x+1}}=\lim_{x\to0}\frac{-x(\sqrt{3x+4}+\sqrt{2x+4})}{x(\sqrt{x+1}+\sqrt{2x+1})}=-\frac{4}{2}=-2$.
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