anonymous
  • anonymous
lim as x -> 0 of (sin x -x)/(tan x - x) using L' Hopitals Rule
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
take derivative of the numerator and denominator independently.
anonymous
  • anonymous
Somehow when I do that I keep getting 0/0 even after taking several derivatives.
goformit100
  • goformit100
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Psymon
  • Psymon
A bit tricky on this one, but when you turn everything into sines and cosines and do l'hopital's once, you can rearrange the equation in a way that it follows these two limit identities: \[\lim_{x \rightarrow 0}\frac{ sinx }{ x }=1 \] \[\lim_{x \rightarrow 0}\frac{ cosx-1 }{ x }=0\]
anonymous
  • anonymous
I'm still little confused how you can rearrange it
Psymon
  • Psymon
|dw:1378875797087:dw| May not be necessary to rearrange it like I did first, but it let me see things better like this. Either way, do l'hopital's rule from here then see if you can move things around to fit the identities above. Should work out.
anonymous
  • anonymous
So is the final the one on the top right?
Psymon
  • Psymon
Yes. Sorry for not making that as clear.
anonymous
  • anonymous
Ok, I got -1/2, is that right?
Psymon
  • Psymon
Nope. Something was missed O.o
anonymous
  • anonymous
Oh geez, this is getting frustrating.
anonymous
  • anonymous
No, -1/2 is right!
Psymon
  • Psymon
Is it? O.o
Psymon
  • Psymon
*smacks calculator*
anonymous
  • anonymous
I suggest apply L'hopital's twice, then rearranging the functions. So as to not deal with the x's.
anonymous
  • anonymous
That's what I did, and I got -1/2
Psymon
  • Psymon
My calculator is stupid then, lol.
anonymous
  • anonymous
Ok Thanks :)
Psymon
  • Psymon
Gotcha, my calculator is just not reliable, lol x_x
Psymon
  • Psymon
@SithsAndGiggles I'm just curious. I do L'hopitals and get the -1/2, but I have my graphing calculator solve it as a limit and graph it and in both instances it gives me 1. Any idea why it might?
anonymous
  • anonymous
I feel the graphing calculator just confuses me even more
anonymous
  • anonymous
Maybe you're computing the limit at infinity?
Psymon
  • Psymon
No, zero, for sure. And then I graphed the function and asked for the value at 0 and also got 1.
Psymon
  • Psymon
Id already gotten myself in the wrong mindset anyway. Once I read that he was still getting 0/0 after several derivatives I didnt even try to do l'hopitals in a normal way x_x Shows how little I trustmyself, lol.

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