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

Textbook Question

Chapter 3, 3.5 - Problem 15b - Calculus of a Single Variable (10th Edition, Ron Larson).
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hkj1385 | (Level 1) Assistant Educator

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

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

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

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scisser | (Level 3) Honors

Posted on

Plug in `oo ` everywhere for x

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

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

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

Use LH's Rule again
`lim_(x->oo)2/2=1 `

Therefore,

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

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