$\text{ The function }f:R-\left\{0\right\}\to R\text{ given by }f\left(x\right)=\frac{1}{x}-\frac{2}{e^{2x}-1}\text{ can be made continuous at x}=0\text{ by }$$\text{ defining }f\left(0\right)\text{ as }$

$\begin{array}{l}:-f\left(0\right)=\underset{x\to 0}{Lt}\frac{1}{x}-\frac{2}{e^{2x}-1}=\underset{x\to 0}{Lt}\frac{e^{2x}-1-2x}{x\left(e^{2x}-1\right)}\\ \text{ =}\underset{x\to 0}{Lt}\frac{2e^{2x}-2}{e^{2x}-1+x.2e^{2x}}\\ \text{ =}\underset{x\to 0}{Lt}\frac{4e^{2x}}{2e^{2x}+2\left[e^{2x}+2x{e}^{2x}\right]}\\ =\frac{4}{2+2}=1\end{array}$