I'm confused with the following problem:
The expression $ \frac{1}{x} $ is clearly not defined at x = 0. However, I read that it could be expressed as a series using the idea $ \lim_{x\to\infty}f(x)=L $ where $ L $ is the "independent term".
More specifically, I am trying to evaluate $ f(z) $ about $ z=0 $ where $ z=\displaystyle\frac{1}{x} $
I've no idea how to start. Could someone please guide me?
The Taylor expansion around 0 of $f(x)$ is: $$ f(x) = f(0) + f'(0) x + \frac{f''(0)}{2!} x^2 + \cdots $$ For this to make sense, all the derivatives at $x = 0$ have to be finite; in particular, $f(0)$ must exist. In your case it doesn't (it and all derivatives are infinite), so there is no Taylor expansion.