## anonymous one year ago limit x tends to 0 for f(x)= (cos(sinx)-cosx)/x^2

1. anonymous

I've tried several things, transformation formulae, series expansions etc etc. Very stuck.

2. anonymous

$\large \lim_{x\rightarrow 0 } \frac{\cos(\sin(x))-\cos(x)}{x^2}$

3. anonymous

On mobile, so thanks for the formatting.

4. Astrophysics

Split it into two limits, one should require squeeze theorem I believe

5. anonymous

Squeeze theorem doesn't make sense if you have a x^2 term I thought?

6. anonymous

No, it does.

7. ganeshie8

we cannot split the limit here because the individual limits don't exist

8. ganeshie8

have u tried taylor ? looks the numerator is $$\mathcal{O}(x^3)$$

9. Astrophysics

I think you maybe right no we can just use L'hopital's rule

10. Astrophysics

0/0

11. imqwerty

what i did - 1) simplified cosA+cosB 2)took the constants out ...... only constant which ws there ws -2 :P 3)applied L hospitality nd then i got (-2)x0 =0 :)

12. Astrophysics

looks good @qwerty!

13. imqwerty

:) thanks @Astrophysics

14. anonymous

cos(A) $$\color{red}{\text{+}}$$ cos(B)?

15. Empty

Yeah I did L'H twice and got 0 as well

16. Astrophysics

Yup did same thing haha

17. anonymous

Oh well, we're required to solve without L'Hospital. So anyone got any clues on that.

18. Empty

Any other restrictions we should know about...?

19. Empty

If you can use series expansions then saying you can't use L'H is practically meaningless imo

20. anonymous

And I don't know what or how Taylor Series are... No other restrictions.

21. ganeshie8

series expansion works nicely

22. ganeshie8

just show that the degree of numerator is at least 3 and you're done

23. Astrophysics

Niceee

24. anonymous

Okay? @ganeshie8 and that's proof cause?

25. ganeshie8

so are you allowed to use below ? $\sin(x)=x-\frac{x^3}{3!} + \frac{x^5}{5!}\mp \cdots$ $\cos(x)=1-\frac{x^2}{2}+\frac{x^4}{4!}\pm\dots$

26. anonymous

yes.

27. ganeshie8

then use them

28. anonymous

I've gone the next few steps... but I don't understand how that gives me a zero.!?!?

29. ganeshie8

$\large \cos(\sin x) = 1-\dfrac{(\sin x)^2}{2}+\cdots = 1 -\dfrac{(x - \frac{x^3}{3!}+\cdots)^2}{2}+\cdots$ fine with above ?

30. anonymous

yes. got that step. Im afraid I need to be led by the hand still though.

31. ganeshie8

\large{\begin{align} &\color{red}{ \cos(\sin x)}-\color{blue}{\cos x}\\~\\ &= \color{red}{1-\dfrac{(x - \frac{x^3}{3!}+\cdots)^2}{2}+\mathcal{O}(x^4)}-\left(\color{blue}{1-\dfrac{x^2}{2}+\mathcal{O}(x^4)}\right) \\~\\ &= \color{red}{-\dfrac{(x - \frac{x^3}{3!}+\cdots)^2}{2}+\mathcal{O}(x^4)}-\left(\color{blue}{-\dfrac{x^2}{2}+\mathcal{O}(x^4)}\right) \\~\\ &= \color{red}{-\dfrac{x^2+\mathcal{O}(x^4)}{2}+\mathcal{O}(x^4)}-\left(\color{blue}{-\dfrac{x^2}{2}+\mathcal{O}(x^4)}\right) \\~\\ &= \color{red}{-\dfrac{\mathcal{O}(x^4)}{2}+\mathcal{O}(x^4)}-\left(\color{blue}{\mathcal{O}(x^4)}\right) \\~\\ &=\mathcal{O}(x^4) \end{align}}

32. ganeshie8

that shows that the degree of each term in the numerator is at least $$4$$

33. anonymous

I'm so sorry, but whats that o like symbol? starting out with calc

34. ganeshie8

If it helps, you may replace $$\mathcal{O}(x^4)$$ by $$x^4(stuff)$$

35. anonymous

got it. thanks

36. anonymous

this actually pretty clever.

37. ganeshie8

\large{\begin{align} &\color{red}{ \cos(\sin x)}-\color{blue}{\cos x}\\~\\ &= \color{red}{1-\dfrac{(x - \frac{x^3}{3!}+\cdots)^2}{2}+x^4(stuff)}-\left(\color{blue}{1-\dfrac{x^2}{2}+x^4(stuff)}\right) \\~\\ &= \color{red}{-\dfrac{(x - \frac{x^3}{3!}+\cdots)^2}{2}+x^4(stuff)}-\left(\color{blue}{-\dfrac{x^2}{2}+x^4(stuff)}\right) \\~\\ &= \color{red}{-\dfrac{x^2+x^4(stuff)}{2}+x^4(stuff)}-\left(\color{blue}{-\dfrac{x^2}{2}+x^4(stuff)}\right) \\~\\ &= \color{red}{-\dfrac{x^4(stuff)}{2}+x^4(stuff)}-\left(\color{blue}{x^4(stuff)}\right) \\~\\ &=x^4(stuff) \end{align}}

38. anonymous

oh no need for that. I worked it out too. thanks a lot lot lot

39. Astrophysics

|dw:1440488705192:dw|

40. anonymous

I'm like that after every single calc question. so many...So many. I'm probably going to keep asking today

41. ganeshie8

|dw:1440489968673:dw|

42. Astrophysics

Cute xD