`lim_(x -> oo) (5 - 2x^(3/2))/(3x - 4)` Find the limit, if possible

Textbook Question

Chapter 3, 3.5 - Problem 17c - Calculus of a Single Variable (10th Edition, Ron Larson).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the limit, hence, you need to replace oo for x in equation:

`lim_(x->oo) (5 - 2x^(3/2))/(3x -4) = (5 - oo)/(3oo - 4) = -(oo)/oo`

Since the result is indeterminate, you need to force `x^(3/2)` and x factors out at numerator and denominator:

`lim_(x->oo) (x^(3/2))(5/(x^(3/2)) - 2)/(x(3 - 4/x) `

Since `lim_(x->oo) 5/(x^(3/2)) = 0` and `lim_(x->oo)4/x = 0,` yields:

`lim_(x->oo) (x^(3/2 - 1))(-2/3) = -2/3*lim_(x->oo) (x^(1/2)) = -2/3*oo = -oo`

Hence, evaluating the given limit yields `lim_(x->oo) (5 - 2x^(3/2))/(3x -4) = -oo.`

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