I am trying to establish the convergence of the following integral: $$ \int_{-\infty}^0 \frac{e^x \sin(x+3)}{\log\vert 2+x \vert} dx $$ I will annotate $\mathcal{U(x_0)}$ when studying the function in the neighborhood of $x_0$
This is what I found: $$ Dom f(x) = (-\infty,-3) \cup (-3,-1) \cup (-1,+\infty)$$ For the interests of the exercise the function is continuos between $ (-\infty,-3) \cup (-3,-1) \cup (-1,0] $
I first notice that in $\mathcal{U(-1)}$ the function is $$ f(x) \sim \frac{C}{\log\vert 1+x+1\vert} \sim \frac{1}{\vert x+1 \vert}$$ therefore not convergent.
Convergence in $\mathcal{U(-3)}$ is also pretty straightforward.
Now I'm having some trouble for $\mathcal{U(-\infty)}$ ; following aid from the book I took the exercise from I tried to estimate it $$ \vert f(x) \vert \le \frac{e^x}{\log \vert x+2\vert} $$ Now the book suggests that it is also $$ \le \frac{1}{x^2} $$ and I can't really understand how it got there.
I suppose it used hierarchies of infinities but I never really developed a good understanding of how to use them in $\mathcal{U(0)}$ and $\mathcal{U(\infty)}$ , are there some tips you could give me?