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Consider
\[f(x)=\frac{(x \log x+x)^{5}(1+2 / x)^{x}}{(x+1 / x)^{5}(\log x+1 / \log x)^{6}}\]

What can we say about $\lim _{x \rightarrow \infty} f(x)$ ?

  1. The function $f(x)$ does not have a limit as $x \rightarrow \infty$
  2. $\lim _{x \rightarrow \infty} f(x)=e^{2}$
  3. $\lim _{x \rightarrow \infty} f(x)=e^{1 / 2}$
  4. $\lim _{x \rightarrow \infty} f(x)=0$
  5. $\lim _{x \rightarrow \infty} f(x)=\infty$

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