Business Applications: Continuous Compounding and the Continuity of Cost Functions

Limits and continuity are not confined to abstract algebra — two of the most practical ideas in business mathematics are built directly on them, and both are regularly tested in Odisha CHSE Business Mathematics & Statistics examinations alongside the purely algebraic limit questions.

Continuous compounding. When a principal $P$ is compounded $n$ times a year at nominal annual rate $r$ for $t$ years, the amount grows to $A = P\left(1 + \dfrac{r}{n}\right)^{nt}$ As the number of compounding periods $n$ is made larger and larger — compounding quarterly, then monthly, then daily, then every hour — interest is credited more and more often, and the amount grows towards a limiting value rather than without bound. Writing $m = n/r$ (so $n = mr$) and letting $n \to \infty$: $A = P\left[\left(1+\dfrac{1}{m}\right)^{m}\right]^{rt} \;\longrightarrow\; P\,e^{rt}$ using the exponential standard limit from the previous section, so that $A = P\,e^{rt}$ is the formula for the amount under continuous compounding — the theoretical limit that ordinary compounding (yearly, monthly, daily…) approaches but never quite exceeds, used routinely in finance to value continuously-accruing interest and growth processes. Worked as Exercise 7 below.

Continuity of cost and revenue functions. Differentiation-based tools used later in this course — marginal cost as the derivative of the total cost function, marginal revenue as the derivative of total revenue — are only meaningful where the underlying function is continuous (indeed, differentiable) at the point in question. Many textbook cost and revenue functions are deliberately modelled as smooth, continuous curves for exactly this reason. …