Is the Coth Approximation Correct for Large x in a QM Textbook?

[intervoxel]

In a QM textbook (Newton), I found the below expression for large x:

<br /> coth(x)\cong 1+2e^{-2x}<br />

I tried

<br /> coth(x)=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=\frac{1+e^{-2x}}{1-e^{-2x}}<br />

 

[George Jones]

Write z = e^{-2x}, so that z is small when x is large. Then,

\coth x = \left(1+z\right)\left(1-z\right)^{-1}.

Do a power series expansion of of this, or do a power series expansion of just \left(1-z\right)^{-1}.

 

[Rajini]

Is it correct to do like this:
for large x; e^x=1+e^x.
So on mere substitution into the eqn.
<br /> coth(x)=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=\frac{1+e^{-2x}}{1-e^{-2x}}<br />
one will get
<br /> coth(x)\cong 1+2e^{-2x}<br />
at low temperature the weighing factor is (1/E).
<br /> x=E/(2k_{\rm B}T)<br />.