Is this function analytic in every point?

Question

Given the function $\ f(x) = \frac{e^{2x} -1-2x}{x} $ , I am asked to determine if there are any non-analytic points, so I managed to find the series for this function, $\sum_{n=1}^∞ \frac{2^{n+1}}{(n+1)!}x^n $ , from what I know every point is analytic

Answer 1

The function $f$ has an isolated singularity at $x_0=0$, this means that there exists an open disk $D(x_0,r)$ for which $f$ is analytic on the punctured disk $D(x_0,r)\backslash \{x_0\}$. What happens is that the singularity is removable

$$ \lim_{x\to 0} f(x) = 0 $$

So, the function is not analytical everywhere

Answer 2

Yes, I am aware that f(x) is not defined on x=0. Yet, I have the example of $ g(x)= \frac{sen(x)}{x} $ being analytical everywhere despite not being defined on x=0, at least according to "Differential equations and boundary value probelms" by Nagle, on page 441, 4th edition, calling such point a removable singularity, so I am a little confused