Use a Numerical approach (i,e., a table of values) to approximate limx→0e2x−1x, Provide support for...

The table for the values of ${\displaystyle \frac{e^{2}x-1}{x}}$
$\begin{array}{|ccccccc|}\hline x& -0.5& -0.05& -0.0001& 0.0001& 0.05& 0.5\\ f(x)& 1.2642& 1.9033& 1.9998& 2.0002& 2.1034& 3.4366\\ \hline\end{array}$
From the table,
Ax x approaches to 0 from left side ${\displaystyle \frac{e^{2}x-1}{x}}$ approaches to 2
As x approaches to 0 from right side ${\displaystyle \frac{e^{2}x-1}{x}}$ approaches to 2
Therefore, ${\displaystyle \underset{x\to 0}{lim}}$ ${\displaystyle \frac{e^{2}x-1}{x}=2}$