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Math Help - Mth101 Q1_8

  1. #1
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    Mth101 Q1_8

    (a) Find the Limit using Lí Hospitalís rule
    lim x→0 (x-tanx)/sinx-x

    (b) show that the given integral converges or not
    ∫+∞e 1/ln3x dx
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by TAHIR View Post
    (a) Find the Limit using L’ Hospital’s rule
    lim x→0 (x-tanx)/sinx-x
    Let L = \lim_{x \to 0} \frac {x - \tan x}{ \sin x - x} ........This goes to \frac {0}{0}

    Apply L'Hopital's, we get:

    L = \lim_{x \to 0} \frac {1 - \sec^2 x}{ \cos x - 1} .......Again, this goes to \frac {0}{0}

    Apply L'Hopital's again, we get:

    L = \lim_{x \to 0} \frac {2 \sec^2 x \tan x}{ \sin x} .......This again goes to \frac {0}{0}, but L'Hopital's would be overkill here, let's just factor

    L = \lim_{x \to 0} \frac {2 \sec^2 x }{ \cos x}

    \Rightarrow L = 2
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