First what did you find the limit is good, correct. Since the limit exists (and finite!) the sequence is bounded (from both sides). One way to show that the given sequence is bounded is using the definition of the limit.
BUT it is desirable to show that it is bounded before you find limit.
In general, if the sequences or function might not be bounded.
As for (1+22/n)^n, it grows very fast but still limited. You can show that it is bounded from above by expanding (1+22/n)^n as binomial expansion:
or alternatively, you can use
Then (1+22/n)^n<=e^22, which basically the limit.