Hi, Kyle, for continuous functions, like trig functions here, the limit of the function is the same as the function applied to the limit. So you find the limit of the algebra expression inside the tan and cot functions and evaluate the trig function.
lim x--> inf tan ( x/x^2+1) = tan (lim x--> ( x/x^2+1))
to evaluate lim x--> x/x^2+1 divide numerator & denominator by x2 so that it is easy to see that the limit is zero.
therefore lim x--> inf tan ( x/x^2+1) = tan(0) = 0
The cot one is similar. The limit inside cot will be pi/4.