Tricky problem on function equation

Question

How do you evaluate $$q(t)=A\cdot (2000t)^t\cdot e^{-2000t}$$ when $q(0)=4000$?

I just can't get around $(2000\cdot0)^0$.

Answer 1

So you want to evaluate $q(t)=A(2000t)^te^{-2000t}$ when $q(0)=4000$?

You have $$q(0)=A(2000\cdot0)^0e^{-2000\cdot0}=A\cdot1\cdot1=A$$

So now using the fact that $q(0)=4000$ you get $$q(0)=A=4000$$ So $A=4000$

Answer 2

The critical observation is the function $f(t)=t^t$, $t>0$, extends continuous to $t=0$, and it could be defined as $f(0)=1$, as $$ \lim_{t\searrow 0}t^t=\exp(t\log t)=\exp (0)=1, $$ since $\lim_{t\searrow 0}t\log t=0.$

Once we know that then as $q(t)=A\cdot 2000^t\cdot t^t \cdot\mathrm{e}^{-2000t}$, then $$ q(0)=A\cdot 1\cdot 1\cdot 1=A. $$ Hence $q(0)=4000$ if and only if $A=4000$.