lim-x-y-x-n-y-n-x-y-

Question Number 12062 by tawa last updated on 10/Apr/17

\lim_{x \to y} \frac{x^{n} - y^{n}}{x - y}

Answered by ajfour last updated on 11/Apr/17

x^{n} - y^{n} = (x - y) (x^{n - 1} + {y x}^{n - 2} +

y^{2} x^{n - 3} + \dots + y^{n - 2} x + y^{n - 1})

\lim_{x \to y} \frac{x^{n} - y^{n}}{x - y} = y^{n - 1} + y . y^{n - 2} +

y^{2} . y^{n - 3} + \dots + y^{n - 2} . y + y^{n - 1}

= n (y^{n - 1}) .

Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 11/Apr/17

L^{^{'}} H o p i t a l r o l e : (y = c o n s t .)

l = \lim_{x \to y} \frac{{n x}^{n - 1}}{1} = {n y}^{n - 1} .