Is this inequality true? $\coth x\leq x^{-1}+x$

Question

By looking at the graph of RHS-LHS, I believe the following inequality holds:

$$\coth x\;\leq\; x^{-1}+x \quad\text{for } x>0$$

I can't think of a way to prove it right now, and I would love a hint and/or reassurance that it's true.

Answer 1

Note that $x+x^{-1}-\coth x$ has derivative $\coth^2 x-x^{-2}\ge 0$, while $\lim_{x\to 0^+}x^{-1}-\coth x=\lim_{x\to 0^+}(-x/3)=0$.