I looked at this question several hours ago and decided to let someone else tackle it because I was unsure. No one has yet added an answer. I am still not sure about this answer, but I will give you something to think about.
When x is large (x2 + 1)/(x + 3) is "close" to x or put another way that rational fraction approaches the line y=x as an asymptote. I don't think it meets the definition of a limit. I certainly cannot give a ε,N type of argument for it.
Since the fraction approaches x, the cot will be "close" to cot x...but again I cannot give any kind of ε argument. However, what may be instructive for you is to simply graph the function on your graphing calculator and look what happens at larger values of x.
I have given you the best help I can. Now I want to ask you a favor. When you take this problem back to class, you will presumably get an answer from your instructor. I would like you to share that answer with me...I am still learning too!