Ernstfalld

2021-09-21

Use a Numerical approach (i,e., a table of values) to approximate $\underset{x\to 0}{lim}\frac{{e}^{2}x-1}{x},$ Provide support for your analysis (slow the table and explain your resoning).

### Answer & Explanation

toroztatG

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

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