Limits

**lehder** · August 12th 2009, 07:11 AM

Hi eveybody,

I find some difiiculties to calculate the following limits:

1)- $\lim_{x\to 1} (2-3x+x^2)tan(\frac{\pi}{2}x)$

2)- $\lim_{x\to \frac{\pi}{4}} \frac{sinx-cosx}{x+xcos4x}$

Can you help me please?

And thank's anyway

**red_dog** · August 12th 2009, 07:26 AM

1) $\lim_{x\to 1}(2-3x+x^2)\tan\frac{\pi x}{2}=\lim_{x\to 1}\frac{(x-1)(x-2)}{\cos\frac{\pi x}{2}}\cdot\sin\frac{\pi x}{2}=$

$=\lim_{x\to 1}\frac{(x-1)(x-2)}{\sin\left(\frac{\pi}{2}-\frac{\pi x}{2}\right)}\cdot\sin\frac{\pi x}{2}=$

$=-\lim_{x\to 1}\frac{1-x}{\sin\frac{\pi(1-x)}{2}}\cdot\lim_{x\to 1}(x-2)\sin\frac{\pi x}{2}=-\frac{2}{\pi}\cdot(-1)=\frac{2}{\pi}$

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