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I've been trying to solve this limit without L'Hospital's rule because I don't know how to use derivates yet. So I tried rationalizing the denominator and numerator but it didn't work.

$$\lim\limits_{x\to 4} \frac{ \sqrt{2x+1}-3 }{ \sqrt{x-2}-\sqrt{2} }$$

By the way, the answer is supposed to be $\frac{2\sqrt{2}}{3}$.

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I'm surprised you've even heard of L'Hopital's rule if you haven't studied derivatives yet. –  David H Apr 6 at 0:44
    
Rationalisation works if you do it for both numerator and denominator; then cancel the common terms. –  Graham Kemp Apr 6 at 1:01

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up vote 1 down vote accepted

$$\lim_{x\to 4} \frac{\sqrt{2x+1}-3}{\sqrt{x-2}-\sqrt{2}} \\ = \lim_{x\to 4} \frac{2(x-4)(\sqrt{x-2}+\sqrt{2})}{(x-4)(\sqrt{2x+1}+3)} \\ = \lim_{x\to 4} \frac{2(\sqrt{x-2}+\sqrt{2})}{(\sqrt{2x+1}+3)} \\ = \frac{2\sqrt 2}{3}$$

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Multiply top and bottom by $\sqrt{2x+1}+3$, and also by $\sqrt{x-2}+2$.

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