Question

The value of $\underset{x\to 1}{lim}{\left(\frac{x^n+x^{n-1}+x^{n-2}+....+x^2+x-n}{x-1}\right)}^{}$ is

• A. $\frac{n(n+1)}{2}$
• B. $0$
• C. $1$
• D. $n$

Answer 1

Answer: (A)

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

$\underset{x\to 1}{lim}\frac{(x^n+x^{n-1}+x^{n-2}+...+x^2+x-n)}{x-1}$

By L' - Hospital

$=\underset{x\to 1}{lim}\frac{n{x}^{n-1}+(n-1)x^{x-2}+...+2x+1}{1}$

$=n+n-1+...+2+1$

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