Question

How would I find this limit analytically? Any hints would be appreciated. :)

Answer 1

\[\lim_{x \rightarrow \frac{\pi}{4}}\left(\frac{1-\tan~\theta}{\sin~\theta-\cos~\theta}\right)\]

Answer 2

umm,,,

Answer 3

@agent0smith

Answer 4

If I divide everything in the denominator by cos\(\theta\) I get tan\(\theta\) - 1

But I don't know if that would help any, mhmm

Answer 5

What do you mean by "analytically"? As in no L'Hopital's rule or series expansions allowed?

Answer 6

\(\Large \lim_{\theta \rightarrow \frac{\pi}{4}}\left(\frac{1-\tan~\theta}{\sin~\theta-\cos~\theta}\right)\)

\(= \Large \lim_{\theta \rightarrow \frac{\pi}{4}} \sec \theta \left(\frac{\cos \theta -\sin~\theta}{\sin~\theta-\cos~\theta}\right)\)

Answer 7

I haven't learned that yet. Analytically here would be factoring the expression so that way we could get rid of the restriction of substituting pi/4 in and finding the limit, mhmm

Answer 8

Oops, it was a typo. x was supposed to be theta

Answer 9

done?

Answer 10

I am not sure what you mean. I see the term analytically as using analysis. You don't really use analysis to find the limit but to prove the limit. i.e. A sequence has a limit \(a\) iff \(\forall \epsilon \mathbb{R} \ \ \exists N\in \mathbb{N} \ \ \forall n\in \mathbb{N} \ \ |x_n-a|<\epsilon\)

Answer 11

me too, mate

Answer 12

There are other questions that ask to prove the limit using the epsilon-delta defintion of the limit and they say so but this question asks to find the limit analytically

Answer 13

\[\lim_{x \rightarrow \frac{\pi}{4}}\left(\frac{1-\tan~\theta}{\sin~\theta-\cos~\theta}\right)\]

\[\lim_{x \rightarrow \frac{\pi}{4}}\left(\sec \theta \cdot \frac{\cos\theta-\sin \theta}{\sin~\theta-\cos~\theta}\right)\]

\[\lim_{x \rightarrow \frac{\pi}{4}}\left(\sec \theta \cdot \frac{\cos\theta-\sin \theta}{-\cos \theta + \sin \theta }\right)\]

\[\lim_{x \rightarrow \frac{\pi}{4}}\left(\sec \theta \cdot -1\right)\]

\[\lim_{x \rightarrow \frac{\pi}{4}}\left(-\sec \theta \right)\]

Answer 14

TSO sorry that will be a bad idea but what may be result than rewrite

(cos theta -sin theta)/(sin theta -cos theta) = - (sin theta -cos theta)/(sin theta -cos theta) = - 1

Answer 15

Yes, thank you Jhonny. It took me a while to realize that :)

Answer 16

was my pleasure so this may be usefully ?

Answer 17

so sec theta = 1/cos theta

cos (pi/4) = 1/sqrt(2)

sec(pi/4) = sqrt(2)

-sec(pi/4) = -sqrt(2)

soooo

\[\lim_{x \rightarrow \frac{\pi}{4}}\left(\frac{1-\tan~\theta}{\sin~\theta-\cos~\theta}\right) =\boxed{\bf{ -\sqrt{2}}}\]

Answer 18

Thanks everyone :)