I am studying cumulative density functions. As an exercise, given a cdf $F$ I am trying to say if functions built on $F$ are also cdf. I would like to check a few answers to understand if my reasoning is correct:
$G(x)=F(bx)$ for some $b<0$ I argued it is not a cdf since $lim_{x\to +\infty} F(bx) = lim_{x\to -\infty}F(x) =0$ but we know that for a cdf $lim_{x\to +\infty}F(x)=1$.
Along the same lines I also said that $G(x)=F(x^2)$ is not a cdf, while $G(x)=F(at), a>0, G(x)=F(x+c), G(x)=F(x^3), G(x)=F(x)^2, G(x)=F(x)^3$ are cdf.
Other functions I checked are $G(x)=1/2(1+tanh(x))$ which seems to be a cdf to me. I have huge problems (I think because of my lack of knowledge in calculus) with the following two: $G(x)=lim_{a\to +\infty} 1/2(1+tanh(ax))$ $G(x)=lim_{a\to 0^+} 1/2(1+tanh(ax))$
How do I calculate $lim_{x\to +\infty}(lim_{a\to +\infty} 1/2(1+tanh(ax)))$ and the other properties as well?
Thanks