x 1 x−1xn−1 is equal to - Vidyadip

MCQ

$\lim x\rightarrow 1x−1xn−1$ is equal to

A
$\frac{\text{n}}{2}$
B
$\frac{\text{n}(\text{n}+1)}{2}$
✓
$n$
D
none of these

✓

Answer

Correct option: C.

$n$

$= {\lim _\limits{\text{x}\to 1}}\frac{{{\text{x}}^{\text{n}}}-1}{\text{x}-1}$
$ =\lim_\limits{\text{x}\to 1}\frac{\left( \text{x}-1 \right)\left( {{\text{x}}^{\text{n}-1}}+{{\text{x}}^{\text{n} -2}}+.....+\text{x}+1 \right)}{\text{x}-1}$
$ =\lim_\limits{\text{x}\to 1}\sum\limits_{\text{i}=0}^{\text{n}-1}{{{\text{x}}^{\text{i}}}}$
$ =\sum\limits_{\text{i}=0}^{\text{n}-1}{{{1}^{\text{i}}}}$
$ =\sum\limits_{\text{i}=0}^{\text{n}-1}{{{1}^{\text{}}}}$
$ = \text{n}$
Hence, this is the answer.

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