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Math Help - limits plz help

  1. #1
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    limits plz help

    1.Find the value of Lim x->0+ (cos x-1)/x^2
    (a)1/4 (b)-2 (c) infinity (d) -1/2 (e) none

    2.find the value of limit lim x->inf x sin(1/x)
    (a)0 (b)1 (c)2 (d)4 (e)none

    3.find the value of limit lim x>infinity x^(2/x)
    (a)2 (b) e (c) ln 2(d)1 (e) none

    4.find the value of lim x->inf ( ln x)^2/x
    (a) - inf (b) -1 (c) 0 (d) e (e) none

    5.find the value of limit lim x->1 (1/(x-1) - 1/ln x)
    (a)-1 (b)0 (c) 1/2 (d) -1/2 (e) none

    6.find the value of limit lim x->pi - (sin x/ 1-cos x)
    (a)1 (b)-1 (c) infinity (d) -infinity (e) none
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  2. #2
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    Quote Originally Posted by bobby77 View Post
    1.Find the value of Lim x->0+ (cos x-1)/x^2
    (a)1/4 (b)-2 (c) infinity (d) -1/2 (e) none
    This helps to use the "extension of the real line" for this problem.
    You have,
    \lim_{x\to\infty}\frac{\cos x-1}{x}\cdot \frac{1}{x}
    The first product exists (it is a standard limit with value of 1). The second does not exist but increases without bound. Thus, the answer is (c)


    2.find the value of limit lim x->inf x sin(1/x)
    (a)0 (b)1 (c)2 (d)4 (e)none
    Let, g=1/x, f=(1/x)\sin x then, the limit of,
    \lim_{x\to\infty}f\circ g can be found by evaluating the inner limit, \lim_{x\to\infty}g=0.So the limit of,
    \lim_{x\to\infty}f\circ g=\lim_{x\to 0}\frac{\sin x}{x} by limit composition rule.

    Quote Originally Posted by bobby77 View Post

    3.find the value of limit lim x>infinity x^(2/x)
    (a)2 (b) e (c) ln 2(d)1 (e) none
    You can express,
    x^{2/x}=\exp (\ln x^{2/x})=\exp \left( \frac{2\ln x}{x} \right)
    The limit of the inner function exists and is zero because any polynomial overtakes a logarithm. Then e^0=1 by the limit composition rule.
    Last edited by ThePerfectHacker; November 7th 2006 at 07:03 AM.
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