1. ## Limits.. help needed!

Hi,

I have a problem in evaluating limits, especially when square roots are involved. I get so far and then hit a wall, although I know there is an answer.

Could anyone please post a step by step breakdown to the following problem, so I can see at what stage I am going wrong.

lim x-->2

sqrt(5x-2) - 2sqrt(2)
--------------------
x^2 -3x +2

Thanks in advance, I've been at this for hours!

2. multiply numerator and denominator by ...

$\displaystyle \sqrt{5x-2} + 2\sqrt{2}$

$\displaystyle \mathop { = \lim }\limits_{x \to 2} \frac{{\sqrt {5x - 2} - 2\sqrt 2 }} {{x^2 - 3x + 2}} \hfill \\$

$\displaystyle {\text{Rationalize the Numerator,}} \hfill \\$

$\displaystyle \mathop { = \lim }\limits_{x \to 2} \frac{{\sqrt {5x - 2} - 2\sqrt 2 }} {{x^2 - 3x + 2}} \times \frac{{\sqrt {5x - 2} + 2\sqrt 2 }} {{\sqrt {5x - 2} + 2\sqrt 2 }} \hfill \\$

Now, finish up.

4. I've tried that, I get...

(5x-2)-8
-------------------------------
(x^2-3x+2)(sqrt(5x-2)+2sqrt(2))

when I plug 2 into x, it gives me

0
---------
(0)(sqrt(8)+2sqrt(2))

which I make 0/0, but I know this cant be right. Any ideas?

5. Originally Posted by kwuk
I've tried that, I get...

(5x-2)-8
-------------------------------
(x^2-3x+2)(sqrt(5x-2)+2sqrt(2))

when I plug 2 into x, it gives me

0
---------
(0)(sqrt(8)+2sqrt(2))

which I make 0/0, but I know this cant be right. Any ideas?
$\displaystyle \mathop { = \lim }\limits_{x \to 2} \frac{{\sqrt {5x - 2} - 2\sqrt 2 }} {{x^2 - 3x + 2}} \hfill \\$

$\displaystyle {\text{Rationalize the Numerator,}} \hfill \\$

$\displaystyle \mathop { = \lim }\limits_{x \to 2} \frac{{\sqrt {5x - 2} - 2\sqrt 2 }} {{x^2 - 3x + 2}} \times \frac{{\sqrt {5x - 2} + 2\sqrt 2 }} {{\sqrt {5x - 2} + 2\sqrt 2 }} \hfill \\$

$\displaystyle \mathop { = \lim }\limits_{x \to 2} \frac{{(5x - 2) - 4(2)}} {{\left( {x^2 - 3x + 2} \right)\left( {\sqrt {5x - 2} + 2\sqrt 2 } \right)}} \hfill \\$

$\displaystyle \mathop { = \lim }\limits_{x \to 2} \frac{{5x - 10}} {{\left( {x - 1} \right)\left( {x - 2} \right)\left( {\sqrt {5x - 2} + 2\sqrt 2 } \right)}} \hfill \\$

$\displaystyle \mathop { = \lim }\limits_{x \to 2} \frac{{5\left( {x - 2} \right)}} {{\left( {x - 1} \right)\left( {x - 2} \right)\left( {\sqrt {5x - 2} + 2\sqrt 2 } \right)}} \hfill \\$

$\displaystyle \mathop { = \lim }\limits_{x \to 2} \frac{5} {{\left( {x - 1} \right)\left( {\sqrt {5x - 2} + 2\sqrt 2 } \right)}} \hfill \\$

Now, finish up. Did you get it Now???

6. Thanks. Helped a lot. My factoring was all messed up.