Calculating a "0/0" limit

**loui1410** · July 10th 2012, 07:04 AM

How can I calculate the following limit without using l'Hôpital's rule?

$\lim_{x\to 2}\frac {\sqrt{x+2}-\sqrt{3x-2}}{\sqrt{4x+1}-\sqrt{5x-1}}$

**Plato** · July 10th 2012, 08:39 AM

Originally Posted by loui1410

How can I calculate the following limit without using l'Hôpital's rule?
$\lim_{x\to 2}\frac {\sqrt{x+2}-\sqrt{3x-2}}{\sqrt{4x+1}-\sqrt{5x-1}}$

The original expression is equal $\left( {\frac{{ - 2x + 4}}{{ - x + 2}}} \right)\left( {\frac{{\sqrt {4x + 1} + \sqrt {5x - 1} }}{{\sqrt {x + 2} + \sqrt {3x - 2} }}} \right).$

**loui1410** · July 10th 2012, 08:49 AM

I got to that and didn't know how to continue.. Only now did I notice that we can simply reduce the fraction

Thanks.

**Plato** · July 10th 2012, 08:59 AM

Originally Posted by loui1410

I got to that and didn't know how to continue.. Only now did I notice that we can simply reduce the fraction

Thanks.

$\left( {\frac{{ - 2x + 4}}{{ - x + 2}}} \right)\left( {\frac{{\sqrt {4x + 1} + \sqrt {5x - 1} }}{{\sqrt {x + 2} + \sqrt {3x - 2} }}} \right)\to(2)\left( {\frac{6}{4} \right)$

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