The Standard Algebraic Limit Formula and Its Equivalent Form

The Same Formula, Written a Different Way

Substitute $x = a(1+t)$ (so $t\to0$ exactly when $x\to a$) into the formula above, and it becomes, after simplification,

$\lim_{t \to 0} \frac{(1+t)^{n}-1}{t} = n$

These are the syllabus's own two named formulae — $\displaystyle\lim_{x\to a}\frac{x^n-a^n}{x-a}=na^{n-1}$ and $\displaystyle\lim_{x\to 0}\frac{(1+x)^n-1}{x}=n$ — and they are different representations of the same underlying result, not two separate facts to memorise independently. The first is stated "around a general point $a$"; the second is the special case obtained by rescaling so the point of interest becomes $0$ and $a$ becomes $1$ (since $a^n$ then equals $1^n=1$ and $a^{n-1}$ equals $1^{n-1}=1$, leaving just the coefficient $n$).