`lim_(x -> -oo) x/(sqrt(x^2 + x))` Find the limit.

Textbook Question

Chapter 3, 3.5 - Problem 28 - 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) x/(sqrt(x^2+x))= (-oo)/ (sqrt(oo+(-oo)))`

Since the result is indeterminate, you need to factor out `x^2` at denominator:

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

`lim_(x->-oo) x/(|x|sqrt (1 + 1/x)) = lim_(x->-oo) x/(-x*sqrt (1 + 1/x))`

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

`lim_(x->-oo) x/(-x*sqrt (1 + 1/x)) = lim_(x->-oo) 1/(-sqrt (1 + 1/x)) = 1/(-1) = -1`

Hence, evaluating the given limit yields `lim_(x->-oo) x/(sqrt(x^2+x)) = -1.`

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