Business Applications of Limits

Limits are not merely an algebraic exercise in the Gujarat Std-12 Statistics (Business Mathematics & Statistics) syllabus — they underpin three genuinely practical results a commerce student meets again in banking, costing, and pricing.

(A) Continuous compounding — deriving $A = Pe^{rt}$

Ordinary compound interest, compounded $n$ times a year at nominal annual rate $r$ for $t$ years, gives the amount:

As the number of compounding periods $n$ grows without bound (daily, then hourly, then every instant — continuous compounding), substitute $m=\dfrac{n}{r}$, so $n=mr$ and $\left(1+\dfrac{r}{n}\right)^{nt}=\left[\left(1+\dfrac{1}{m}\right)^{m}\right]^{rt}$. As $n\to\infty$, $m\to\infty$ too, and the standard limit $\lim_{m\to\infty}\left(1+\frac{1}{m}\right)^{m}=e$ (Section 3) gives:

the continuous-compounding formula used by banks and in every subsequent finance numerical in this course.

(B) Average Fixed Cost tending to zero — the "spreading overhead" effect

If a firm's total fixed cost is $F$ (rent, salaries, machinery — costs that do not change with output), the average fixed cost per unit at output $x$ is $\text{AFC}(x)=\dfrac{F}{x}$. As output expands without limit:

— overhead gets spread over more and more units, so the fixed-cost burden per unit shrinks toward (but never quite reaches) zero. This is the mathematical reason large-scale producers can profitably sell at a lower price per unit than small-scale ones.

(C) Discontinuity in slab-based pricing …