Yeah. Though the defenders of transfinite (ordinal and cardinal) numbers do in fact assert that there are many infinite numbers, such as aleph zero or omega. They are just usually somewhat embarrassed about this and therefore only talk about "ordinals" or "cardinals". It's like trying to hide that you drink beer by saying you merely drink lagers and ales.
Absense of limit is not always an infinity, e.g. for sequence (-1)^n.
Even if sequence is unbounded, it does not always converge to infinity, e.g. n^((1+(-1)^n)/2): 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 12, 1, 14, 1, 16, 1, 18, 1, 20 ...
Convergence of sequence x(n) to infinity by definition is: for each real number ε>0 there exists a natural number N(ε) such that for every number n≥N(ε) we have |x(n)|>ε.