`lim_(x -> oo) (5x^3 + 1)/(10x^3 - 3x^2 + 7)` Find the limit.

Textbook Question

Chapter 3, 3.5 - Problem 24 - Calculus of a Single Variable (10th Edition, Ron Larson).
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hkj1385's profile pic

hkj1385 | (Level 1) Assistant Educator

Posted on

`limx->oo (5x^3 + 1)/(10x^3 - 3x^2 + 7)`

`or, lim x->oo {((5x^3)/x^3)+(1/x^3)}/{((10x^3)/x^3)-((3x^2)/x^3) + (7/x^3)`

`or, limx->oo {5 + (1/x^3)}/{10-(3/x)+(7/x^3)} = (5+0)/(10-0+0) = 5/10 = 1/2`

``

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scisser | (Level 3) Honors

Posted on

Plug in ` ` `oo ` everywhere for x

`lim_(x->oo) (5(oo)^3+1)/(10(oo)^3-3(oo)^2+7)=oo/oo `

` `

Since you have ` ` `oo/oo ` , you can use L^Hopital's Rule and differentiate the numerator and denominator independently.

`lim_(x->oo)(15x^2)/(30x^2-6x)=oo/oo `

Use LH's Rule again

`lim_(x->oo)(30x)/(60x-6)=oo/oo `

Once again, use LH's Rule

`lim_(x->oo)30/60=1/2 `

Therefore,

the `lim_(x->oo)(5x^3+1)/(10x^3-3x^2+7)=1/2 `

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