Calculating population with calculus

Question

A major corporation is building a 4325 acre complex of homes, offices, stores, schools, and churches in the rural community of Glen Cove. As a result of this development, the planners have estimated that Glen Clove's population (in thousands) t yr from now will be given by the following function.

 

(45t^2+125t+100)/ (t^2+2t+10)

 

What is the current population?

 

What will be the population in the long run?

 

Any explanation would be appreciated! 

Answer 1

(45t²+125t+100)/ (t²+2t+10)

What is the current population?  Current time is t=0.  Plug in t=0 to get the answer.

What will be the population in the long run?  When t is large (the long run), the t² terms dominate:

 

limit as t→∞ P(t) = limit as t→∞ (45t²/t²) = 45 thousand