Question

The value of $\mathop {\lim }\limits_{x \to 1} \dfrac{{x + {x^2} + ...... + {x^n} - n}}{{x - 1}}$ is
$

{\text{a}}{\text{. n}} \\
{\text{b}}{\text{. }}\dfrac{{n + 1}}{2} \\
{\text{c}}{\text{. }}\dfrac{{n\left( {n + 1} \right)}}{2} \\
{\text{d}}{\text{. }}\dfrac{{n\left( {n - 1} \right)}}{2} \\
$

Answer 1

Hint: - Apply L’ Hospital’s Rule.

Given limit is
$\mathop {\lim }\limits_{x \to 1} \dfrac{{x + {x^2} + ...... + {x^n} - n}}{{x - 1}}$
Put$\left( {x = 1} \right)$, in this limit
$\Rightarrow \dfrac{{1 + 1 + 1 + .................. + 1 - n}}{{1 - 1}}$
As we know sum of 1 up to n terms is equal to n.
$\Rightarrow \dfrac{{n - n}}{{1 - 1}} = \dfrac{0}{0}$
So, at$x = 1$, the limit is in form of $\dfrac{0}{0}$
So, apply L’ Hospital’s rule
So, differentiate numerator and denominator separately w.r.t.$x$
As we know differentiation of${{\text{x}}^n} = n{x^{n - 1}}$, and differentiation of constant term is zero.
$\Rightarrow \mathop {\lim }\limits_{x \to 1} \dfrac{{\dfrac{d}{{dx}}\left( {x + {x^2} + ...... + {x^n} - n} \right)}}{{\dfrac{d}{{dx}}\left( {x - 1} \right)}} \Rightarrow \mathop {\lim }\limits_{x \to 1} \dfrac{{1 + 2x + 3{x^2} + ........ + n{x^{n - 1}} - 0}}{{1 - 0}}$
Now, put$x = 1$,$\Rightarrow \dfrac{{1 + 2 + 3 + ...................... + n}}{1}$
$\Rightarrow 1 + 2 + 3 + ...................... + n = \sum\limits_{r = 1}^n r$
Now as we know sum of first natural numbers is $\left( {{\text{i}}{\text{.e}}{\text{.}}\sum\limits_{r = 1}^n r = \dfrac{{n\left( {n + 1} \right)}}{2}} \right)$
$\mathop {\lim }\limits_{x \to 1} \dfrac{{x + {x^2} + ...... + {x^n} - n}}{{x - 1}} = \sum\limits_{r = 1}^n r = \dfrac{{n\left( {n + 1} \right)}}{2}$
Hence option (c) is the correct answer .

Note: - In such types of questions the key concept we have to remember is that, whenever the limit comes in the form of $\dfrac{0}{0}$ always apply L’ hospital’s rule, (i.e. differentiate numerator and denominator separately), and always remember the sum of first natural numbers then we will get the required answer.