Assume that \(f\) is differentiable at \(a\). Evaluate \[\lim_{x\to a}\frac{a^nf(x)-x^nf(a)}{x-a},\quad n\in\mathbb{N}.\]
Solution
1. Rewrite the numerator: \(a^n f(x) - x^n f(a) = a^n(f(x) - f(a)) - f(a)(x^n - a^n)\).
\(a^n f(x) - x^n f(a) = a^n f(x) - a^n f(a) + a^n f(a) - x^n f(a) = a^n(f(x) - f(a)) - f(a)(x^n - a^n)\).
2. \(f\) is differentiable at \(a\), so \(\frac{f(x) - f(a)}{x - a} \to f'(a)\) as \(x \to a\).
definition of the derivative.
3. \(\frac{x^n - a^n}{x - a} = x^{n-1} + x^{n-2}a + \cdots + a^{n-1} \to n a^{n-1}\) as \(x \to a\).
factorization of \(x^n - a^n\); or the derivative of \(x \mapsto x^n\) at \(a\).
4. Hence \[\lim_{x \to a}\frac{a^n f(x) - x^n f(a)}{x - a} = a^n f'(a) - f(a)\,(n a^{n-1}) = a^n f'(a) - n a^{n-1} f(a).\] Proof: 1 splits the quotient as \(a^n\frac{f(x) - f(a)}{x-a} - f(a)\frac{x^n - a^n}{x-a}\), and 2, 3 give the two limits.