`lim_(x -> oo) (5x^(3/2))/(4x^(3/2) + 1)` Find the limit, if possible

Textbook Question

Chapter 3, 3.5 - Problem 18b - 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) (5x^(3/2))/(4x^(3/2) + 1) = (5oo)/(4oo + 1) = (oo)/oo`

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

`lim_(x->oo) (x^(3/2))(5)/(x^(3/2)(4 + 1/(x^(3/2))))`

Since `lim_(x->oo) 1/(x^(3/2)) = 0` , yields:

`lim_(x->oo) (x^(3/2-3/2))*(5/4)= 5/4*lim_(x->oo) (x^0) = 5/4*1 =5/4`

Hence, evaluating the given limit yields `lim_(x->oo) (5x^(3/2))/(4x^(3/2) + 1) = 5/4.`

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