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I am having trouble with a simple question : Consider $\bar{R}$ the extended real line and $ 0 < q < \infty$. Let $x_n $a sequence in $\bar{R}$ with $x_n \geq 0, \forall \ n $. Suppose that $x_n \rightarrow a$ where $a = + \infty$ What I can say about the limit below? $lim_{n \rightarrow + \infty} (x_n)^{\frac{-1}{q}}$ |
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HINT: Let $p=\dfrac1q$; clearly $0<p<\infty$. For sufficiently large $n$, $x_n\ne 0$, and $x_n^{-p}=\dfrac1{x_n^p}$ actually makes sense. Now what is $\lim\limits_{n\to\infty}x_n^p$? |
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