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Math Help - challenging limits

  1. #1
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    challenging limits

    can someone help me find the limit of

    1. lim ((sin x)/ (pi -x)) as x to pi
    2. lim ((x+1)ln x/ sin x) as x tend to 0 from above
    3. lim xe^(1/x) as x tends to infinity.
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  2. #2
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    Question

    Are you allowed to use l'Hospital's Rule?
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by alexandrabel90 View Post
    can someone help me find the limit of

    1. lim ((sin x)/ (pi -x)) as x to pi
    2. lim ((x+1)ln x/ sin x) as x tend to 0 from above
    3. lim xe^(1/x) as x tends to infinity.
    I think "challenging" may be pushing it.

    1.Notice that \lim_{x\to\pi}\frac{\sin(x)}{\pi-x}=-\lim_{x\to\pi}\frac{\sin(x)-\sin(\pi)}{x-\pi}=-\bigg[\sin(x)\bigg]'\bigg|_{x=\pi}=1

    2. \lim_{x\to0}\frac{\ln(x)(x+1)}{\sin(x)}. Merely note that on a small enough neighborhood of 0 this is clearly bigger than \ln(x) (or is it?...what can you say?)

    3. \lim_{x\to\infty} xe^{\frac{1}{x}}=\lim_{x\to0^+}\frac{e^x}{x}\to\in  fty
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