Question

The value of $\underset{n\to \infty }{\lim }\left[\frac{x^n}{x^n+1}\right]$, where $x<-1$ is

• A. $\frac{1}{2}$
• B. $-\frac{1}{2}$
• C. $1$
• D. None of these

Answer 1

The correct option is C

$1$

Explanation for the correct option:

Given, $\underset{n\to \infty }{\lim }\left[\frac{x^n}{x^n+1}\right]$

Solve the given expression as follows:

$$\begin{gathered} \underset{n\to \infty }{\lim }\left[\frac{x^n}{x^n+1}\right]=\underset{n\to \infty }{\lim }\left[\frac{1}{1+{\displaystyle \frac{1}{x^n}}}\right] \\ \Rightarrow \underset{n\to \infty }{\lim }\left[\frac{x^n}{x^n+1}\right]=\left[\frac{1}{1+0}\right] \\ \Rightarrow \underset{n\to \infty }{\lim }\left[\frac{x^n}{x^n+1}\right]=1 \end{gathered}$$

Therefore, The value of $\underset{n\to \infty }{\lim }\left[\frac{x^n}{x^n+1}\right]$, where $x<-1$ is $1$.

Hence, option C is the correct answer.