Answer
$\displaystyle \lim_{t\rightarrow+\infty}E(t)=\infty,\qquad\lim_{t\rightarrow+\infty}\frac{E(t)}{I(t)}=0.4$
In the long term, according to the model, U.S. exports to China will rise without bound.
In the long term, according to the model, U.S. exports to China will be be $ 40\%$ of the value of the U.S. imports from China.
The problem, in reality, is that imports and exports can not rise without bound. This tells us that the models should not be extrapolated too far into the future.
Work Step by Step
Using theorem 10.2, $f(t)=\displaystyle \frac{polynomial}{polynomial}$,
we can calculate the limit of $f(t)$ as $ t\rightarrow\pm\infty$ by ignoring all powers of $t$ except the highest power in both the numerator and denominator.
(Functions of the form $f(t)=\displaystyle \frac{polynomial}{1}$ also satisfy the terms of the theorem.)
$\displaystyle \lim_{t\rightarrow+\infty}E(t)=\lim_{t\rightarrow+\infty}0.4t^{2}\qquad$
which has the form
$0.4(Big\ positive)^{k}=Big$
$\displaystyle \lim_{t\rightarrow+\infty}E(t)=\infty$
In the long term, according to the model, U.S. exports to China will rise without bound.
$\displaystyle \lim_{t\rightarrow+\infty}\frac{E(t)}{I(t)}=\lim_{t\rightarrow+\infty}\frac{0.4t^{2}}{t^{2}}=\lim_{t\rightarrow+\infty}\frac{0.4}{1}=0.4$
In the long term, according to the model, U.S. exports to China will be be $ 40\%$ of the value of the U.S. imports from China.
The problem, in reality, is that imports and exports can not rise without bound. This tells us that the models should not be extrapolated too far into the future.
