`lim_(x -> -oo) sqrt(x^4 - 1)/(x^3 - 1)` Find the limit.

Textbook Question

Chapter 3, 3.5 - Problem 32 - 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) (sqrt(x^4-1))/(x^3-1) = (sqrt(oo^4-1))/((-oo)^3-1)= (oo)/(-oo)`

Since the result is indeterminate, you need to factor out `x^4` at numerator and `x^3` at denominator:

`lim_(x->-oo) (sqrt (x^4(1 - 1/x^4)))/(x^3(1-1/x^3)) =`

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

Since `lim_(x->-oo) 1/(x^3) = 0` and `lim_(x->oo) 1/x^4 = 0` , yields:

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

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

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