Consider the function

$f(x)=\frac{e^{2x}-1}{x}.$

$($a$)$ Calculate $\underset{x0}{lim}f(x).$

$($b$)$ Use a Maclaurin series to rewrite $f(x)$ so that it can be calculated without loss of significance for $x$ close to $0.$ Express your answer as an infinite sum.

$($c$)$ Truncate the infinite sum in Part $($b$)$ to a finite sum, such that the truncated terms are less than ${10}^{-10}$ at $x=0\mathrm{.}001.$