Hello, I know I posted this question recently but I wanted to update with my progress. I have figured out what the limit should be but I would really appreciate help with how to use the definition of the limit of a sequence to prove it! What I have is:
Suppose n is extremely large, then both n^2+1 and n^2 have almost the same size. Similarly both n and n+1 have almost the same size. Therefore, as n grows larger and larger, we can replace the numerator by sqrt (n^2)=n and the denominator by n. Therefore the guess value for the limit should be 1 since the numerator and denominator should cancel.
I am pretty sure this is right, but I am not sure how to answer it in the way this question is asking.