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Theorem for Any Positive Integer n

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theorem

Theorem - For any positive integer n,
`lim_(x -> a) (x^n - a^n)/ (x - a)` = ` na^(n - 1)`

The expression in the above theorem for the limit is true even if n is any rational number and a is positive.

Proof :  Dividing `(x^n – a^n)` by (x – a), we see that

`x^n – a^n = (x–a) (x^(n–1) + x^(n–2) a + x^(n–3) a^2 + ... + x a^(n–2) + a^(n–1))`

Thus ,

`lim_(x-> a) (x^n - a^n)/(x - a) = lim_(x -> a) (x^(n-1) + x^(n-2) a + x^(n-3) a^2 + ... + x a^(n - 2) + a^(n - 1))`

`= a^(n – l) + a a^(n–2) +. . . + a^(n–2) (a) +a^(n–l)`

`= a^(n–1) + a^(n – 1) +...+a^(n–1) + a^(n–1)`  (n terms)

`= na ^(n - 1)` .

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