Difficulty in finding a counterexample
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 $x\to\infty$.
Difficulty in finding a counterexample
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 $x\to\infty$.