`lim_(x -> -oo) (2x)/(x^6 - 1)^(1/3)` Find the limit.

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Chapter 3, 3.5 - Problem 34 - 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 expression under the limit:

`lim_(x_>-oo) (2x)/(root(3)(x^6 - 1)) = -oo/oo`

Since the limit is indeterminate, you need to force factor `x^6` out to denominator:

`lim_(x_>-oo) (2x)/(x^2*root(3)(1 - 1/x^6)) = lim_(x_>-oo) (2)/(x*root(3)(1 - 1/x^6))`

Since `lim_(x_>-oo) 1/x^6 = 0` yields:

`lim_(x_>-oo) (2)/(x*root(3)(1 - 1/x^6)) = lim_(x_>-oo) (2)/(x) = 2/(-oo) = 0`

Hence, evaluating the given limit, yields `lim_(x_>-oo) (2x)/(root(3)(x^6 - 1)) = 0.`

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