Limit of a cotangent function as x approaches infinity?

Question

Is the limit well defined?
Why or why not?

the limit as x → ∞ of Cot ((x^2+1)/(x+3))

Answer 1

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!