lim_(x→0) (((x+1)^(n+1) −(n+1)(x+1)+n)/x^2 ) n∈N Calculate. (Without a L′Hopital′s rule)


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Question Number 125292 by Kurbanklichevs last updated on 09/Dec/20

$\lim_{x \to 0} \frac{{(x + 1)}^{n + 1} - (n + 1) (x + 1) + n}{x^{2}} n \in N$ $C a l c u l a t e . (W i t h o u t a L^{'} {H o p i t a l}^{'} s r u l e)$
	Answered by Dwaipayan Shikari last updated on 09/Dec/20

	$\lim_{x \to 0} \frac{{(1 + x)}^{n + 1} - (n + 1) (x + 1) + n}{x^{2}}$ $\lim_{x \to 0} \frac{1 + (n + 1) x + \frac{n (n + 1)}{2!} x^{2} - (n + 1) (x + 1) + n}{x^{2}}$ $\lim_{x \to 0} \frac{1 - n - 1 - n + \frac{n (n + 1)}{2!} x^{2}}{x^{2}}$ $= \frac{n (n + 1)}{2}$
	Commented by Kurbanklichevs last updated on 10/Dec/20

	$T a n k y o u$
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