I'm wonderig if the next is true in general,
$\lim\limits_{x\to\ a} f(x)\pm\lim\limits_{y\to\ a} g(y)= \lim\limits_{x\to\ a} f(x)\pm\ g(x).$
The question arises because of Cauchy's principal value, in which I have an integral say for example $\int_{-\infty }^{\infty} \sin(x) dx $ and taking the limits as $\int_{-a }^{0} \sin(x) dx +\int_{0 }^{b} \sin(x) dx $ a and b tend to infinity, they both diverge, but if you take $\lim\limits_{a\to\infty}\int_{-a}^{a} \sin(x) dx = 0 $ because sine is an odd function. wouldn't it be correct to just take the VP always instead of the other definition, I mean what's the point of separating the literals
thanks