Mathematical issue(s)

Hi fellow board members. I’m having exam on Monday in functional analysis in which I suprisingly often fail. So I thought of ways to efficiently learn the subject but the problem I encounter with almost every task is how/where to start it. So if you can resolve following tasks and describe your entire process of thinking I’d be greatly thankful and hopefully, I won’t fail on Monday. First of all, the topic is finding the drivatives and limits of various functions.
I hope you can understand the notation I’m using, due to the limit of board notation options
Task no.1:
Evaluate the limits of the following functions:
lim[x->0]((x*ctg(x)-1)/x)
lim[x->0]((x*e^x+x-2*e^x+2)/x^3)
lim[x->0](a^x-a^(sin(x)))/x^3)
lim[x->1](2-x)^(tg((Pi*x)/2))
lim[x->infinity](x^(1/x))
Task no.2:
Prove that, if |x|!=1, the equation
0.5*(arctg((2*x)/1-x^2))=arctg(x) + C(x) is true,
witch C(x) defined as:
           Pi/2, if -infinty<x<-1,
C(x)={0, if -1<x<1,
           -Pi/2, if 1<x<infinity
I won’t bother you with any additional tasks, I’d be suprised if you solve even one of these. But anyway, thanks for your help.
Answer #1
I would suggest learning from your math books instead. Having other people solve these problems for you won’t help you. You need to practice this yourself and should go for direct feedback from a teacher or professor if you don’t know where to start.
Answer #2
Answer #3
it’s unfortunate, but LSky is right.
You should be discussing this with your professor, thats the first thing I would have done