amonoconnor
  • amonoconnor
I'm stuck on this one... I just don't see how to manipulate or substitute to evaluate it without getting 0. Find the Limit of: *mathematical expression in first comment" Thank you! Any and all help is greatly appreciated!
Calculus1
jamiebookeater
  • jamiebookeater
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amonoconnor
  • amonoconnor
\[\lim_{x \rightarrow \pi/4}\frac{ 1- \tan x }{\sin x - \cos x}\]
amonoconnor
  • amonoconnor
Sorry about the goofy fraction there, I retyped it twice, and the error didn't go away.
zepdrix
  • zepdrix
\[\large\rm \frac{1-\color{orangered}{\tan x}}{\sin x-\cos x}=\frac{1-\color{orangered}{\frac{\sin x}{\cos x}}}{\sin x-\cos x}\]Let's multiply top and bottom by cos(x) to get rid of that fraction up there.

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zepdrix
  • zepdrix
\[\large\rm =\frac{\cos x-\cos x\frac{\sin x}{\cos x}}{(\sin x-\cos x)\cos x}\]
zepdrix
  • zepdrix
\[\large\rm =\frac{\cos x-\sin x}{(\sin x-\cos x)\cos x}\]Can you see where this is heading? :)
Melissa_Something
  • Melissa_Something
I dont, Jesus. But good luck!!! <3 xD
amonoconnor
  • amonoconnor
I understand everything up to here, but.. uhh, where do I go now? :/
zepdrix
  • zepdrix
Hmmm, let's factor a -1 out of each term in the numerator. It might click in your brain when we do :)
zepdrix
  • zepdrix
\[\large\rm =\frac{-(\sin x-\cos x)}{(\sin x-\cos x)\cos x}\]
zepdrix
  • zepdrix
Any ideas? Hmmmmm...
amonoconnor
  • amonoconnor
.... and the -1 just goes in front of the "lim" part?
zepdrix
  • zepdrix
Ya, it can go where ever you want it to go :)
amonoconnor
  • amonoconnor
Why don't we switch the signs in the denominator as well?
zepdrix
  • zepdrix
You mean when we factored out the -1? Or you're suggesting we should follow that up with another factoring of -1 in the bottom? I'm just a lil confused by the question >.<
amonoconnor
  • amonoconnor
Yes, like, when you said we factor a -1 out of the top... why aren't we just taking it out of the whole fraction/problem, switching signs in the bottom to make it cosx(cosx - sinx)? Is it because -1 is technically = -1/+1?
zepdrix
  • zepdrix
Yes, good observation. If we did it to the whole thing, it would be more like -1/-1 being taken out, which is really no change at all.
zepdrix
  • zepdrix
I'm gonna color something and hopefully it jumps out at you as to why we took a -1 out of the numerator,\[\large\rm =\frac{-(\color{orangered}{\sin x-\cos x})}{(\color{orangered}{\sin x-\cos x})\cos x}\]
amonoconnor
  • amonoconnor
I follow now.. I think I get it. So, now that I have this ... (hold on)
amonoconnor
  • amonoconnor
\[-1*\lim_{x \rightarrow \pi/4}\frac{1}{\cos(x)}\]
amonoconnor
  • amonoconnor
now that I have this, I just plug in pi/4?
zepdrix
  • zepdrix
So you were able to divide out "the problem piece", nice!
amonoconnor
  • amonoconnor
= \[(\sqrt{2})/2\]
zepdrix
  • zepdrix
Hmm close. We're getting sqrt(2)/2, but that's in the denominator. And also, don't forget about your negative outside of the limit.
amonoconnor
  • amonoconnor
right, so the "final answer" is...
amonoconnor
  • amonoconnor
\[-\frac{2}{\sqrt{2}}\]
amonoconnor
  • amonoconnor
?
zepdrix
  • zepdrix
Good, rationalize or simplify though
amonoconnor
  • amonoconnor
Ahhh... touche. I was wondering how my book had just negative sqrt.2... Got it, multiply top and bottom by sqrt.2, cancel out the 2's , and we get -sqrt.2!! :D
zepdrix
  • zepdrix
wooo, nice job broski!
amonoconnor
  • amonoconnor
Thanks again zepdrix! It's been a while, but your help is as helpful as ever!!

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