The increase in a person’s body temperature  t(t), above 98.6ºf, can be modeled by the function  t(t)=4t/t^2+1, where  trepresents time elapsed. what is the meaning of the horizontal asymptote for this function?

The meaning of the horizontal asymptote is that the increase in a person's body temperature is a constant value when long time passes.

You can obtain that value by calculating the limit of the function when t → ∞:

                 4t
T(t) =  => Lim T(t) when t → +∞ = 0.
             t^2 + 1

That means that at long term the Temperature tends to remain at 98.6°F.