What is the end behavior of y=1/x+2

1/ ( x + 2)

The degree of the polynomial in the  numerator  is less than  the  degree of the polynomial in the denominator....thus...we will have a horizontal asymptote  at  y = 0 .....the behavior at  both ends  will  approach  this asymptote y  = 0

See the graph here :  https://www.desmos.com/calculator/s6j0qch9li