I need to find an upper bound for $f: \mathbb{R}\setminus \{0\} \to \mathbb{R}$ where $f$ is defined by $$f(x):= \sqrt{\frac{2x+e^{-2x}-1 }{4x^2}}-\sqrt{\frac{1-4xe^{-2x}-e^{-4x} }{4x^2(1-e^{-2x})}}$$ for each $x \in \mathbb{R}\setminus \{0\}$.

My attempt

Let $x \in \mathbb{R}\setminus \{0\}$. Then we have that

\begin{align} f(x)&\leq \sqrt{ \frac{2x+e^{-2x}-1 }{4x^2} - \frac{1-4xe^{-2x}-e^{-4x} }{4x^2(1-e^{-2x})}}\\ &=\sqrt{\frac{x(1+e^{-2x})-(1-e^{-2x})}{2x^2(1-e^{-2x})}}. \end{align}

I don't know how I can continue. Can you help me, please?

  • $\begingroup$ do you have any especification on the range that the upper bound should hold and what kind of bound you want? The question is bit too generic. Eg, $|f(x)| + 1$ is a stupid upper bound $\endgroup$ – Daniel Aug 13 at 16:05
  • $\begingroup$ For large and positive $x$ the function should be really close to $\frac{\sqrt{2x-1} - 1}{2x}$. You can use calculus to estimate how close $\endgroup$ – Daniel Aug 13 at 16:09

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