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Question:

thomasstudent
thomasstudent
Student
Community / Jr. College

calculate lim ((x(sin x)^1/2) - x^1/2 tan x)) / (x^4 x^1/2) as x->0

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Posted by thomasstudent on Saturday November 7, 2009 at 8:32 PM and tagged with limit, math.

Answers:

  1. neela
    neela Teacher
    Graduate School

    eNotes Editor

    To calculate the limit of  {x(sinx)^(1/2)-x^(1/2)*tanx}/x^4*x^(1/2)} as x approaches zero.

    Solution:

    x has to approach zero only from positive. The expression is imaginary for negative values of x.So we consider the limit from positive x approaching zero.

    Let f(x) = {x(sinx)^(1/2)-x^(1/2)*tanx}/x^4*x^(1/2)}

    =(sinx)^(1/2)/x^(3.5) - tanx/x^4.

    We use Maclaurin's expansion of sin x and tan x.

    Sinx =x-x^3/3!+x^5/5!-x^7/7!+.....

    Tanx= x+x^3/3+2x^5/15+17x^7/315+.....

    Therefore,

    f(x)=(1/x^3.5){x-x^3/6+x^5/120 -....}^(1/2) -

    (1/x^4){x+x^3/3+(2x^5/15)+17x^7/315)+....}

    <x/x^3.5 - x/x^4

    =1/x^2.5-1/x^3

    =x^2.5(x^.5-1)/x^5.5

    =(x^0.5-1)/x^3

    =- infinity as x approaches 0 from right.

    f(x) is imaginary when x is -ve.

    For the existence of a limit, right limit and left limit should exist and be equal. And this condition is not satisfied by f(x).Hence there is no limit as x approaches zero.

     

     

     

    Rate answer:

    Posted by neela on Sunday November 8, 2009 at 4:04 AM