lim-x-0-x-1-n-1-n-1-x-1-n-x-2-n-N-Calculate-Without-a-L-Hopital-s-rule-

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

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