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Math Help - limits

  1. #1
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    limits

    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
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  2. #2
    MHF Contributor red_dog's Avatar
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    Quote Originally Posted by Dili View Post
    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}
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  3. #3
    MHF Contributor red_dog's Avatar
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    Quote Originally Posted by Dili View Post
    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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