Question

Prove that \(\lim\limits_{x \to a}\frac{x^n - a^n}{x - a} = na^{n - 1}\) for all values of n

Answer 1

Case – 1 : Let n be a positive integer. 

xⁿ – aⁿ = (x – a)(x^{n -1} + x^n-2 . a + x^n-3 . a² + ……. + a^n-1

Case – 2: Let n be a positive integer 

Put n = -m, m > 0

Case – 3: Let n = p/q where p and q are integers and q ≠ 0