Question

If $n\in N,$ and $\left[x\right]$ denotes the greatest integer less than or equal to x, then ${lim}_{x\to n}(-1{)}^{\left[x\right]}$ is equal to

• A. $1$
• B. $-1$
• C. $0$
• D. None of the above

Answer 1

Answer: (D)

None of the above

We have,
$LHL=\underset{x\to n^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(n-h)=\underset{h\to 0}{lim}{(-1)}^{\left[\right(n-h\left)\right]}=\underset{h\to 0}{lim}(-1{)}^{(n-1)}=(-1{)}^{(n-1)}$
$RHL=\underset{x\to n^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(n+h)=\underset{h\to 0}{lim}{(-1)}^{\left[\right(n+h\left)\right]}=\underset{h\to 0}{lim}(-1{)}^n=(-1{)}^n$
$\therefore \underset{x\to n^{-}}{lim}(-1{)}^{\left[x\right]}\ne \underset{x\to n^{+}}{lim}(-1{)}^{\left[x\right]}$
So, $\underset{x\to n}{lim}(-1{)}^{\left[x\right]}$ does not exist.