Confusion with calculating a limit $F(x):=\int_{x}^{2x}e^{-2t}t^{-1}dt$

The question is given in the following way:

Define $F:(0,\infty)\rightarrow \mathbf{R}$ by $$F(x):=\int_{x}^{2x}e^{-2t}t^{-1}dt$$ Determine whether or not $\lim_{x\rightarrow 0^+}F(x)$ exists.

In the first step I write the series of $e^{-2t}$ :$$e^{-2t}=1-(2t)+\frac{(2t)^2}{2!}-...$$Then $$\frac{e^{-2t}}{t}=\frac{1}{t}-2+\frac{2^2t}{2!}- ...$$Then I integrate the series termwise $$\int_{x}^{2x}e^{-2t}t^{-1}dt=\int_{x}^{2x}\frac{1}{t}dt-\int_{x}^{2x}2dt+...$$So, if we take the $\lim_{x\rightarrow0^+}$ and integrating termwise I get $ln(2).$

Process-wise, it seems right, but many things are coming into my mind, such as how the infinite terms are valid for termwise integration and how we are interchanging the limit. Please help me to write it freshly. Thank you beforehand!!!