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Over the real line, prove that

$\lim_{x\rightarrow \infty} \left (\frac{x+\log 9}{x-\log 9} \right )^{x}=81$

I've tried L Hospital Rule, which gets too complicated and I'm trying by Squeeze theorem now.

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    $\begingroup$ Do you know that $(1+(a/x))^{x}\to e^a$ as $x\to\infty$. Try to guess $a$ in your question. $\endgroup$ – Paramanand Singh Nov 7 '18 at 16:37
  • $\begingroup$ @ParamanandSingh $\lim_{(x-\log 9)\rightarrow \infty}\left ( 1+ \frac{2\log9}{x-\log 9} \right )^{\log 9}\left ( 1+ \frac{2\log9}{x-\log 9} \right )^{x-\log 9}=1.e^{2\log9}=81$ $\endgroup$ – Akash Gaur Nov 8 '18 at 11:49
  • $\begingroup$ Yes that's exactly what you need to do. $\endgroup$ – Paramanand Singh Nov 8 '18 at 12:15
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Hint: $$\frac{x+a}{x-a}=1+\frac{2a}{x-a}$$

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Hint:

Let $u=x-\log 9$. As $x \to \infty$ we have $u \to \infty$.

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