## dinainjune Group Title lim┬(x→π/2)⁡〖(π(π-2x) tan⁡〖(x-π/2)〗)/(2(x-π) 〖cos〗^2 x)〗 3 years ago 3 years ago

1. dinainjune Group Title

this is limit of trygonometry, i put it on the ms.word

2. malevolence19 Group Title

Thats indeterminate (0/0) so you would do l'hospital's rule. Have you done that?

3. dinainjune Group Title

yes, i have done. but, i'm stuck in trygonometry. can you do that?

4. malevolence19 Group Title

The functions involved in the limit are arbitrary as long as they are easily differentiable and the limits actually exist. So taking the derivatives you have: $\frac{\pi}{2}\lim_{x \rightarrow \frac{\pi}{2}}\frac{(-2)(\tan(x-\frac{\pi}{2}))+(\pi-2x)\sec(x-\frac{\pi}{2})\tan(x-\frac{\pi}{2})}{\cos^2(x)+(x-\pi)(2)(-\sin(x))\cos(x))}$ However you notice this is still undeterminate. (0/0) so you have to differentiate it again.

5. malevolence19 Group Title

So differentiate it again: $\frac{\pi}{2}\lim_{x \rightarrow \frac{\pi}{2}}\frac{-2 \pi \csc^3(x) (2 \sin(x)+(\pi-2 x) \cos(x))}{4 (\pi-x) \cos(2 x)-4 \sin(2 x)}$ Evaluating this should give you 2.

6. dinainjune Group Title

Thank you

7. malevolence19 Group Title

No problem, sorry its so gross looking :/

8. dinainjune Group Title

it's okay, you help me anyway :)