Taking the limit as x approaches infinity of: can be rewritten as:
The limit as x approaches infinity of: MINUS the limit as x approaches infinity of:
But this isn't true.
If I re-arrange algebraically in the first example, I will get x=2.
However, if I solve the rewritten problem, I get infinity minus infinity or undefined...??
I sense your confusion stems from language barrier. if you say lim(f(x)) as x-->n, exist means at the limit f(n) is finite, or can be resolve without anbguity. if your expression leads to (inf) + (inf), that limit cannot be resolve. You will have to rearrange your expression to an equivalent form. if now the limit is finite, then that limit exists, but if it is still infinte, than you say the limit does not exist. Hopefully this helps.
There is a theorem that says "IF exists and exists then .
You may be thinking that " " means that the limit exists and is infinity. That is not the case. Saying that " " means "the limit does NOT exist" (in a particular way).