mathop {lim }limits_{x to 1} frac{{x + {x^2} | vedclass

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Question

(C) We can rewrite the expression as: $\mathop {\lim }\limits_{x 	o 1} \left( {\frac{{x - 1}}{{x - 1}} + \frac{{{x^2} - 1}}{{x - 1}} + \dots + \frac{

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$\frac{{n(n + 1)}}{2}$

Vedclass · Answer

(C) We can rewrite the expression as: $\mathop {\lim }\limits_{x 	o 1} \left( {\frac{{x - 1}}{{x - 1}} + \frac{{{x^2} - 1}}{{x - 1}} + \dots + \frac{{{x^n} - 1}}{{x - 1}}} ight)$ Using the formula $\mathop {\lim }\limits_{x 	o a} \frac{{{x^k} - {a^k}}}{{x - a}} = k{a^{k - 1}}$,for $a = 1$: $\mathop {\lim }\limits_{x 	o 1} \frac{{{x^k} - 1}}{{x - 1}} = k(1)^{k - 1} = k$ Thus,the sum becomes: $1 + 2 + 3 + \dots + n = \frac{{n(n + 1)}}{2}$

$\mathop {\lim }\limits_{x \to 1} \frac{{x + {x^2} + ...... + {x^n} - n}}{{x - 1}}$ is equal to

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