Limits and Derivatives — MATHS STD 11 Science — Question

3. n

Solution:

$= {\lim _\limits{\text{x}\to 1}}\frac{{{\text{x}}^{\text{n}}}-1}{\text{x}-1}$

$ =\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.