Standard Limits

Certain limits recur so often in business mathematics — particularly once differentiation is introduced in the next chapter — that it is worth learning their values as standard results, rather than re-deriving each one from scratch every time.

1. The power limit. For any positive integer (more generally, any rational) $n$: $\lim_{x \to a} \dfrac{x^n - a^n}{x - a} = n\,a^{n-1}$ This is the factorisation method applied once and for all to the general case $x^n - a^n = (x-a)(x^{n-1}+x^{n-2}a+\cdots+a^{n-1})$, a sum of exactly $n$ terms, each tending to $a^{n-1}$ as $x \to a$.
2. The sine limit. With $x$ measured in radians: $\lim_{x \to 0} \dfrac{\sin x}{x} = 1$ This is the foundation of every trigonometric-function result used later in differentiation, and it holds only when the angle is in radians, never in degrees.
3. The exponential (compounding) limit. $\lim_{n \to \infty} \left(1 + \dfrac{1}{n}\right)^{n} = e$ where $e \approx 2.71828$ is Euler's number. Unlike the other standard limits, this one is not evaluated by substitution or factorisation at all — it is, in effect, the definition of the number $e$, arrived at by watching the expression settle down as $n$ grows without bound (illustrated in Worked Example 6). This particular limit is the mathematical backbone of continuous compounding, a business application covered later in this chapter. …