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

Textbook Question

Chapter 3, 3.5 - Problem 15a - Calculus of a Single Variable (10th Edition, Ron Larson).
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hkj1385's profile pic

hkj1385 | (Level 1) Assistant Educator

Posted on

`lim x->oo (x^2 + 2)/(x^3 - 1)`

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

`or, limx->oo {(1/x)+(2/x^3)}/{1-(1/x^3)}`

`or, lim x->oo{(1/x)+(2/x^3)}/{1-(1/x^3)} = (0+0)/(1-0) = 0`

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scisser | (Level 3) Honors

Posted on

Plug in `oo/oo ` everywhere you have x

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

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

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

Use LH's Rule again

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

2 divided by a really big number is 0.

Therefore,

the `lim_(x->oo)(x^2+2)/(x^3-1)=0 `

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