## abannavong 3 years ago i need help with finding the limit if x-->0 sinx/x

1. satellite73

answer or method? answer is 1 and it is best just to memorize it, you will use it repeatedly

2. abannavong

the method! plz!

3. zzr0ck3r

do you know l'hopital rule yet?

4. satellite73

if you want a proof that does not use derivatives, i.e. a geometric argument, it should be in any intro calc book

5. zzr0ck3r

nm silly question

6. abannavong

i dont think i do yet

7. satellite73

it is a pain to do without using calculus, so best idea is to look it up with calculus it is almost a triviality an nice graph will show it though http://www.wolframalpha.com/input/?i=sin%28x%29%2Fx

8. zzr0ck3r

I agrea.. If you must know, google it. Or just wait a few weeks then you can do it in your head

9. zzr0ck3r

agree*

10. LolWolf

We have that: $\sin(x)\le x\le\tan(x)=\frac{\sin(x)}{\cos(x)}$We find the limit as $$x\to 0$$ of this, and use the squeeze theorem to show the following: Since $$\sin(x)\ne 0$$, for this case: $\frac{\sin(x)}{\sin(x)}\le\frac{x}{\sin(x)}\le\frac{\frac{\sin(x)}{\cos(x)}}{\sin(x)}=\frac{1}{\cos(x)}$We cancel the $$\sin(x)$$ in the first term: $1\le\frac{x}{\sin(x)}\le \frac{1}{\cos(x)}$Since $$x\to 0$$, we simply plug in $$x=0$$ for the last expression: $1\le\frac{x}{\sin(x)}\le \frac{1}{1}=1$By the squeeze theorem, then: $\lim_{x\to 0}\frac{x}{\sin(x)}=1$ We wish to find: $\lim_{x\to 0}\frac{\sin(x)}{x}=\frac{1}{\lim_{x\to 0}\frac{x}{\sin(x)}}=\frac{1}{1}=1$Thus, we are done.

11. abannavong

kk!

12. abannavong

i need help with finding another limit!

13. abannavong

s|dw:1347246353846:dw|

14. satellite73

this one is much easier rationalize the numerator by multiplying top and bottom by $\sqrt{2+x}+\sqrt{2}$

15. satellite73

you will get $\frac{2+x-2}{x(\sqrt{x+2}+\sqrt{2})}$ cancel and get $\frac{1}{\sqrt{x+2}+\sqrt{2}}$ take the limit by replacing $$x$$ by 0

16. abannavong

oh ok! i get it!

17. satellite73

great

18. abannavong

thank you!

19. satellite73

yw

20. abannavong

i actually need help again lolz

21. abannavong

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22. LolWolf

It's 0.