1. anonymous

$\lim_{x \rightarrow 2}(\sqrt{x+2}-\sqrt{2x})\div(x^2-2x)$

2. anonymous

@phi

3. welshfella

direct substitution gives 0/0 i would apply l'hopitals rule

4. anonymous

5. anonymous

this week we just learned about intro to limits

6. phi

one thought is to multiply top and bottom by $$\sqrt{x+2}+ \sqrt{2x}$$

7. welshfella

ok so we need to find different way

8. anonymous

let me multiply it

9. phi

the top is a difference of squares

10. anonymous

|dw:1438036441844:dw|

11. phi

and don't multiply out the bottom you should get $\frac{ (x+2) - 2x} { x(x-2)(\sqrt{x+2}+\sqrt{2x})}$

12. phi

simplify the top

13. anonymous

@phi did you multiply the whole equation by the conjugate?

14. anonymous

or by what factor?

15. phi

the top was (a-b)(a+b) = a^2 - b^2 where a and b are the square roots the bottom , I just show the multiplication

16. phi

I can't tell what you did , but it looks too complicated to be right.

17. phi

should I go slower?

18. anonymous

yes, please. I get where you said that they're difference of squares. But I don't get which factor did you multiply the whole equation to get that result

19. phi

$\frac{(\sqrt{x+2}-\sqrt{2x})}{(x^2-2x)} \\\frac{(\sqrt{x+2}-\sqrt{2x})}{x(x-2)}$ (factored the bottom, hopefully something cancels...

20. phi

$\frac{(\sqrt{x+2}-\sqrt{2x})}{x(x-2)} \cdot \frac{(\sqrt{x+2}+\sqrt{2x})}{(\sqrt{x+2}+\sqrt{2x})}$

21. phi

it is easier to see the pattern if we call the first term a and the second b then we have (a-b)(a+b) which we know (right?) is a^2 - b^2

22. anonymous

yes

23. phi

or you can FOIL it out. either way the top becomes $(\sqrt{x+2})^2 - ( \sqrt{2x})^2 \\ x+2 - 2x \\ 2-x$ or -(x-2)

24. anonymous

okay, I got this part.

25. phi

so we now have the limit of $\frac{-(x-2)}{x(x-2)(\sqrt{x+2}+\sqrt{2x})}$

26. phi

as long as x is not *exactly* 2 , we can cancel (x-2) from top and bottom then we can take the limit

27. anonymous

-1/8 right?

28. phi

yes

29. anonymous

OMG, thank you so much. I really learned everything

30. phi

yw