## appleduardo 3 years ago whats the limit of the following function?

1. appleduardo

$\frac{ x ^{2}-\sqrt{x} }{ 1-\sqrt{x} }$

2. appleduardo

x-->1

3. zordoloom

Do you know how to solve this?

4. shubhamsrg

you may try rationalizing, both numerator and denom separately..

5. appleduardo

but how can i do that? do first the denomitador and then de numerator or both at the same time?

6. shubhamsrg

doesnt matter..

7. shubhamsrg

any order you prefer..

8. appleduardo

??

9. zordoloom

The limit is -3.

10. shubhamsrg

you may rationalize either denom or nume first,,order wont matter..

11. sauravshakya

Use L'Hospital Rule

12. appleduardo

mm but how? maybe something like this? $\frac{ x ^{2} +\sqrt{x} }{ x ^{2} +\sqrt{x} }$ multiply the original function times the rationalization of the numerator?

13. godfreysown

i get 3x

14. appleduardo

yeep, i know the limit, but what i dont, is how to find it

15. zordoloom

Just take the derivative of the top, and then take the derivative of the bottom,

16. sauravshakya

It is -3

17. zordoloom

Then plug in 1 for x and that will give you the limit.

18. appleduardo

mm but how can i find the limit without using L'hopital? cos my teacher doesnt allow me to do that yet, he want me to find the limit algebraically

19. appleduardo

:/

20. zordoloom

Just multiply top and bottom by conjugates then.

21. appleduardo

respectively? i mean the numerator conjugate times the original numerator and the denominator conjugate times the original denominator?

22. zordoloom

no,

23. zordoloom

What you want to do is rationalize one side.

24. zordoloom

So, just multiply top and bottom by 1+sqrt(x)

25. zordoloom

You should get 1-x for the bottom

26. appleduardo

$\frac{ x ^{2}-\sqrt{x}(1+\sqrt{x}) }{1-x }$ so thts my result after doing what u said :p is it correct?

27. Aussie

you would have parenthesis around x^2 and sqrt x

28. appleduardo

mm yes so what can i do next?

29. Aussie

I am not sure actually you still have on the denominator 1-1 which is still dividing by a zero

30. Aussie

31. hartnn

$$\huge\frac{ (x ^{2}-\sqrt{x})(1+\sqrt{x}) }{1-x }\times \frac{x^2+\sqrt x}{x^2+\sqrt x}$$

32. hartnn

x^4-x = x(x^3-1) = x(x-1)(x^2x+x+1)

33. hartnn

put x=1 after cancelling 1-x