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

Textbook Question

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

Since the result is indeterminate, you need to force `x^(3/2)` and `x^(1/2) ` factors at numerator and denominator:

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

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

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

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

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