Question

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!

Answer 1

\[\lim_{x \rightarrow \pi/4}\frac{ 1- \tan x }{\sin x - \cos x}\]

Answer 2

Sorry about the goofy fraction there, I retyped it twice, and the error didn't go away.

Answer 3

\[\large\rm \frac{1-\color{orangered}{\tan x}}{\sin x-\cos x}=\frac{1-\color{orangered}{\frac{\sin x}{\cos x}}}{\sin x-\cos x}\]Let's multiply top and bottom by cos(x) to get rid of that fraction up there.

Answer 4

\[\large\rm =\frac{\cos x-\cos x\frac{\sin x}{\cos x}}{(\sin x-\cos x)\cos x}\]

Answer 5

\[\large\rm =\frac{\cos x-\sin x}{(\sin x-\cos x)\cos x}\]Can you see where this is heading? :)

Answer 6

I dont, Jesus. But good luck!!! <3 xD

Answer 7

I understand everything up to here, but.. uhh, where do I go now? :/

Answer 8

Hmmm, let's factor a -1 out of each term in the numerator.
It might click in your brain when we do :)

Answer 9

\[\large\rm =\frac{-(\sin x-\cos x)}{(\sin x-\cos x)\cos x}\]

Answer 10

Any ideas?
Hmmmmm...

Answer 11

.... and the -1 just goes in front of the "lim" part?

Answer 12

Ya, it can go where ever you want it to go :)

Answer 13

Why don't we switch the signs in the denominator as well?

Answer 14

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 >.<

Answer 15

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?

Answer 16

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.

Answer 17

I'm gonna color something and hopefully it jumps out at you as to why we took a -1 out of the numerator,\[\large\rm =\frac{-(\color{orangered}{\sin x-\cos x})}{(\color{orangered}{\sin x-\cos x})\cos x}\]

Answer 18

I follow now.. I think I get it. So, now that I have this ... (hold on)

Answer 19

\[-1*\lim_{x \rightarrow \pi/4}\frac{1}{\cos(x)}\]

Answer 20

now that I have this, I just plug in pi/4?

Answer 21

So you were able to divide out "the problem piece",
nice!

Answer 22

= \[(\sqrt{2})/2\]

Answer 23

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.

Answer 24

right, so the "final answer" is...

Answer 25

\[-\frac{2}{\sqrt{2}}\]

Answer 26

?

Answer 27

Good, rationalize or simplify though

Answer 28

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

Answer 29

wooo, nice job broski!

Answer 30

Thanks again zepdrix! It's been a while, but your help is as helpful as ever!!