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Question Number 125089 by bounhome last updated on 08/Dec/20

$${lim}_{x\to -\mathrm{\infty }}(\sqrt{x^2+3x}-x)=?$$

Answered by Bird last updated on 08/Dec/20

$$f\left(x\right)=\sqrt{x^2+3x}-x\Rightarrow$$$$f\left(x\right)=\mid x\mid \sqrt{1+\frac{3}{x}}-x$$$$forx<0f\left(x\right)=-x\sqrt{1+\frac{3}{x}}-x$$$$\sim -x(1+\frac{3}{2x})-x=-2x-\frac{3}{2}$$$$\Rightarrow {lim}_{x\to -\mathrm{\infty }}f\left(x\right)=+\mathrm{\infty }$$

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