<s>[INST] limit of $\sqrt{(x^2+1)/(x^3+1)}$ as $x$ approaches negative infinity $\lim_{x \rightarrow - \infty }  \sqrt{ \frac{x^2+1}{x^3+1} } $
My teacher says that no limit exists, but Wolfram Alpha says the limit is 0. I'm confused.
Any helps are welcome.
 [/INST] The answer depends on the underlying domain of the expression. To see this, you may write
$$\sqrt{ \frac{x^2+1}{x^3+1} } \stackrel{x<-1}{=} \sqrt{-1} \cdot \frac{1}{\sqrt{|x|}} \cdot \sqrt{ \frac{1+\frac{1}{x^2}}{1+\frac{1}{x^3}} } = \begin{cases} \mbox{not defined} & \mbox{ in } \mathbb{R} \\ \pm i \cdot \frac{1}{\sqrt{|x|}} \cdot \sqrt{ \frac{1+\frac{1}{x^2}}{1+\frac{1}{x^3}} } \stackrel{x \to -\infty}{\longrightarrow} 0 & \mbox{ in } \mathbb{C}\end{cases}$$
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