I have the function y=3x+1/x+1

Since the highest power on x in both the numerator and denominator is the same, as x increases without bound in both directions.......the function  will  approach the ratio of the coefficients on both these powers of x.....so  3/1 = 3

So....the function approaches y = 3 in both directions

See the graph here : https://www.desmos.com/calculator/rc5bgtop7n