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Math Help - about one limit

  1. #1
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    about one limit

    Hello,

    How to solve the following question?

    lim_{x\rightarrow1^{-}}(2+cos(\pi x))^{tg(\frac{\pi x}{2})}

    Thanks in advance.
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  2. #2
    Junior Member mathemanyak's Avatar
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    lim_{x\rightarrow1^{-}}(2+cos(\pi x))^{tg(\frac{\pi x}{2})}



    we know tan.cot=1, so we multiply and divides this eqn. by 2+cos(\pi x)^{cot(\frac{\pi x}{2})} then you will find your solution.
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  3. #3
    MHF Contributor red_dog's Avatar
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    \displaystyle\lim_{x\nearrow 1}(2+\cos\pi x)^{\tan\frac{\pi x}{2}}=\lim_{x\nearrow 1}\left[(1+1+\cos\pi x)^{\displaystyle\frac{1}{1+\cos\pi x}}\right]^{\displaystyle(1+\cos\pi x)\tan\frac{\pi x}{2}}=

    \displaystyle=e^{\displaystyle\lim_{x\nearrow 1}2\cos^2\frac{\pi x}{2}\tan\frac{\pi x}{2}}=e^{\displaystyle\lim_{x\nearrow 1}\sin\pi x}=e^0=1
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  4. #4
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    Quote Originally Posted by mathemanyak View Post
    lim_{x\rightarrow1^{-}}(2+cos(\pi x))^{tg(\frac{\pi x}{2})}



    we know tan.cot=1, so we multiply and divides this eqn. by 2+cos(\pi x)^{cot(\frac{\pi x}{2})} then you will find your solution.
    How would that help? (2+ cos(\pi x))^a(2+ cos(\pi x))^b= (2+ cos(\pi x))^{a+b}, not (2+ cos(\pi x))^{ab}.
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