12.4 Real-World Applications

12.4 Real-World Applications

Limits of exponential and rational models describe long-term behaviour in growth/decay, finance, economics, and astronomy.

Example 14 (Radioactive decay)

$A(t) = A_0 e^{-kt}$. As $t \to \infty$, $e^{-kt} \to 0$, so $\lim_{t \to \infty} A(t) = 0$.

Example 15 (Continuous compounding)

$A(t) = P_0 e^{rt}$ ($r > 0$). As $t \to \infty$, $A(t) \to \infty$ — the investment grows without bound.

Example 16 (Demand)

$D(p) = \dfrac{100}{p+1}$: $\lim_{p \to \infty} D(p) = 0$ — demand vanishes as price becomes arbitrarily large.

Example 17 (Astronomy)

Brightness $B(d) = \dfrac{L}{d^2}$: $\lim_{d \to \infty} B(d) = 0$ — apparent brightness tends to zero with distance.