I have a problem with my homework:

1. prove only by the defenition of the limit that $\displaystyle

\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \frac{1}{{\cos x}} \ne \infty

$

Thanks ahead

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- Apr 4th 2009, 12:54 AMomertHelp with limits
I have a problem with my homework:

1. prove only by the defenition of the limit that $\displaystyle

\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \frac{1}{{\cos x}} \ne \infty

$

Thanks ahead - Apr 4th 2009, 03:41 AMProve It
Are you allowed to use L'Hospital's Rule?

If so, it's very easy...

$\displaystyle \lim_{x \to \frac{\pi}{2}} \frac{1}{\cos{x}} = \lim_{x \to \frac{\pi}{2}} \frac{\frac{d}{dx}(1)}{\frac{d}{dx}(\cos{x})}$

$\displaystyle = \lim_{x \to \frac{\pi}{2}} \frac{0}{-\sin{x}}$

$\displaystyle = \frac{0}{-\sin{\frac{\pi}{2}}}$

$\displaystyle = \frac{0}{-1}$

$\displaystyle = 0$.

Without L'Hospital, not so easy...

All I can think of is the Sandwich Theorem, but I can't think of what to sandwich it inside.

I know that $\displaystyle \frac{\sin{x}}{x}\leq \frac{1}{\cos{x}}$.

We just need to find something that $\displaystyle \frac{1}{\cos{x}}$ is less than or equal to... - Apr 4th 2009, 03:46 AMmr fantastic
- Apr 4th 2009, 03:59 AMProve It
- Apr 4th 2009, 04:37 AMomert
I can't use any theorem. I need to prove it only by the defenition. let be M>0 so the any delta>0.... x-pi/2 -->>> 1/cosx<=M

thanks ahead - Apr 4th 2009, 04:38 AMProve It
- Apr 4th 2009, 06:06 AMmatheagle
You need to approach $\displaystyle \pi/2$ from the right and the left.

- Apr 4th 2009, 06:42 AMTwig
Like matheagle says, approach from left and right.

$\displaystyle \lim_{x\rightarrow \frac{\pi}{2}} \frac{1}{cos x} = \infty $

And from the left:

$\displaystyle \lim_{x\rightarrow \frac{-\pi}{2}} \frac{1}{cos x} = -\infty $

I hope this is correct.. - Apr 4th 2009, 06:54 AMmatheagle
- Apr 4th 2009, 06:57 AMTwig
oops yeah, thx (Cool)