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

Textbook Question

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

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

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

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

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

Hence, evaluating the given limit yields `lim_(x->oo) (5x^(3/2))/(4x^2 + 1) = 0.`

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