limits

**Dili** · September 4th 2007, 09:37 AM

Can you help me to evaluate the following limits

lim(x->0+) ((x^0.5)+lnx)/(2tanx + cot^(-1)x)

lim(x->pi/4) (tanx)^tan2x

**red_dog** · September 4th 2007, 10:03 AM

Originally Posted by Dili

lim(x->pi/4) (tanx)^tan2x

$\displaystyle\lim_{x\to\frac{\pi}{4}}(\tan x)^{\tan 2x}=\lim_{x\to\frac{\pi}{4}}\left[(1+\tan x-1)^{\frac{1}{\tan x-1}}\right]^{(\tan x-1)\tan 2x}=$
$\displaystyle=e^{\displaystyle \lim_{x\to\frac{\pi}{4}}(\tan x-1)\frac{2\tan x}{1-\tan^2x}}=\displaystyle e^{\displaystyle -\lim_{x\to\frac{\pi}{4}}\frac{2\tan x}{1+\tan x}}=e^{-1}$

**red_dog** · September 4th 2007, 10:18 AM

Originally Posted by Dili

lim(x->0+) ((x^0.5)+lnx)/(2tanx + cot^(-1)x)

L'Hospital:
$\displaystyle\lim_{x\searrow 0}\frac{\sqrt{x}+\ln x}{2\tan x+\cot^{-1} x}=\lim_{x\searrow 0}\displaystyle\frac{\frac{1}{2\sqrt{x}}+\frac{1}{ x}}{\frac{2}{\cos^2x}-\frac{1}{1+x^2}}=\frac{\infty}{1}=\infty$

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