## anonymous 3 years ago Evaluate the limit, if it exists lim h-0 ((1/(x+h)^2 - 1/x^2 ))/ h

1. myininaya

Try combining top fractions. And get rid of the compound fraction. Unless you already know derivatives then we can actually skip this part.

2. anonymous

so a common denominator for the top fractions? do i expand the (x+h)^2

3. myininaya

You will need to.

4. anonymous

so x^2 + 2xh + h^2, what would be the common denominator

5. myininaya

x^2(x+h)^2

6. myininaya

|dw:1379983226588:dw|

7. myininaya

|dw:1379983241382:dw|

8. anonymous

2xh + h^2/ x^2 (x+h)^2 x (1/h)

9. anonymous

can i factor out a h from the numerator?

10. anonymous

so i factored out an h from the numerator, than cancelled the h from 1/h

11. myininaya

What happen to your negative in from 2xh?

12. myininaya

h/h=1 yep yep

13. anonymous

i have lim h--0 2x+h/ x^2(x+h)^2

14. anonymous

so now i can "sub" in 0 into h, but what value goes for x?

15. myininaya

Well yeah but still you are missing a negative on top

16. myininaya

only h is going somewhere x stays

17. myininaya

It said what happens as h goes to zero (it said nothing for x)

18. myininaya

Did you figure out what negative you are missing?

19. anonymous

ok i think i see my mistake so i went thru the stages again and right now I have lim h--0 -2x-h/x^2 (x+h)^2

20. anonymous

because I factored out an h, but im confused as to my next step

21. myininaya

h goes to 0

22. anonymous

is my answer suppose to be just a number? or will it have a value and x

23. myininaya

24. anonymous

-2x-0/x^2 (x+0) ^2 ?

25. myininaya

(-2x-0)/(x^2(x+0)^2) yes Simplify.

26. anonymous

-2/x?

27. anonymous

not rlly syre how to simplify the denominator

28. mathslover

$$\cfrac{(-2x-0)}{(x^2)(x+0)^2}$$ (Just for ease - wrote the LaTeX)

29. myininaya

-2x-0=-2x x^2(x+0)^2=x^2(x)^2 (since x+0=x) Use law of exponents.

30. myininaya

Recall one of the laws of exponents a^m * a^n=a^(m+n)

31. anonymous

ok so my answer is -2/x^3

32. myininaya

yep

33. anonymous

thank you :)

34. mathslover

But the denominator should be x^4 @myin

35. myininaya

-2x/x^4=-2/x^3

36. anonymous

I have another question: find the limit if it exists limx-- -2 2-|x|/ 2+x

37. anonymous

thats lim x approaches -2

38. anonymous

can I just sub in -2 for x? so that makes it 0/0

39. mathslover

Oh sorry @myininaya - didn't saw that :) Thanks !

40. myininaya

$\lim_{x \rightarrow -2} \frac{2-|x|}{2+x}$ ?

41. anonymous

yes

42. myininaya

So we can't just simply plug in -2 because our denominator will be 0. Is there someway to make the function continuous at x=-2 so we can just plug in Well |x|=x if x>=0 and -x if x<0 Since x approaches -2 then x is negative since we are looking at what surrounds a negative number so we have |x|=-x

43. myininaya