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    1. Evaluate the following limits.

    (a) lim x^2/(cos9x-cos5x) =_______
    x->0

    (b) lim x^(1/3) ln x =_______
    x->0+

    (c) lim ((1/ln x) - (1/x-1)) =_________
    x->1+

    (d) lim ((1 + (5/x))^(x/2) =__________
    x->+infiniti

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    https://brainmass.com/math/basic-algebra/simple-calculus-question-384248

    Solution Preview

    1.

    lim_(x->0) x^2/(cos(9 x)-cos(5 x))

    Indeterminate form of type 0/0. Applying L'Hospital's rule we have, lim_(x->0) x^2/(-cos(5 x)+cos(9 x)) = lim_(x->0) (( dx^2)/( dx))/(( d(-cos(5 x)+cos(9 x)))/( dx)):
    = lim_(x->0) (2 x)/(5 sin(5 x)-9 sin(9 x))

    Factor out constants:
    = 2 (lim_(x->0) x/(5 sin(5 x)-9 sin(9 x)))

    Indeterminate form of type 0/0. Applying L'Hospital's rule we have, lim_(x->0) x/(5 sin(5 x)-9 sin(9 x)) = lim_(x->0) (( dx)/( dx))/(( d(5 sin(5 x)-9 sin(9 x)))/( dx)):
    = 2 (lim_(x->0) 1/(25 cos(5 x)-81 cos(9 x)))

    The limit of a quotient is the quotient of the limits:
    = 2/(lim_(x->0) (25 cos(5 x)-81 cos(9 x)))

    The limit of a sum is the sum of the limits:
    = 2/(25 (lim_(x->0) cos(5 x))-81 (lim_(x->0) cos(9 x)))

    Using the continuity of cos(x) at x = 0 write lim_(x->0) cos(5 x) as cos(lim_(x->0) 5 x):
    = 2/(25 cos(lim_(x->0) 5 x)-81 (lim_(x->0) cos(9 x)))

    Using the continuity of cos(x) at x = 0 write lim_(x->0) cos(9 x) as cos(lim_(x->0) 9 x):
    = 2/(25 cos(lim_(x->0) 5 x)-81 cos(lim_(x->0) 9 x))

    Factor out constants:
    = 2/(25 cos(5 (lim_(x->0) x))-81 cos(lim_(x->0) 9 x))

    The limit of x as x approaches 0 is 0:
    = 2/(25 cos(0)-81 cos(lim_(x->0) 9 x))

    Factor out constants:
    = 2/(25 cos(0)-81 cos(9 (lim_(x->0) x)))

    The limit of x as x approaches 0 is 0:
    = -1/28

    2.

    lim_(x->0) x^(1/3) log(x)

    Indeterminate form of type 0·infinity. Let t = 1/x, then lim_(x->0) x^(1/3) log(x) = lim_(t->infinity) (1/t)^(1/3) log(1/t):
    = lim_(t->infinity) (1/t)^(1/3) log(1/t)

    Indeterminate form of type 0·infinity, write lim_(t->infinity) (1/t)^(1/3) ...

    Solution Summary

    simple calculus question

    $2.49

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