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

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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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Transform the equation under lim, `(2x+1)/sqrt(x^2-x)` .

Divide the numerator and denominator by |x|, take into account that `x^2=|x|^2`:

`(2x+1)/sqrt(x^2-x) = (2*x/|x| + 1/|x|)/sqrt(1-x/|x|^2)`

When `x->-oo,` `1/|x| -> 0`, `x/|x|^2 -> 0`  and x/|x|->-1.
So the limit is `(2*(-1)+0)/sqrt(1-0)` = -2.