Difficulty in finding a counterexample I am finding difficulties in finding a counterexample to the following statement. If $f\colon (0,\infty) \to(0,\infty) $ is uniformly continuous, then
$$\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1.$$

Let $f(x)= e^{-x^2}$. Then 
$$ f(x+\tfrac1x)=e^{-x^2-2-x^{-2}}=f(x)\cdot e^{-2}\cdot e^{-x^{-2}}$$so that the quotient tends to $e^{-2}$ as $x\to\infty$.
