Question

$\underset{x\to 0}{\lim }{(-1)}^{\left[x\right]}$, where $\left[x\right]$ denotes the greatest integer function

• A. $0$
• B. $1$
• C. $-1$
• D. Does not exist

Answer 1

The correct option is D

Does not exist

Explanation for the correct option:

Step 1: Expanding the given equation and applying the limits

$\underset{x\to 0}{\lim }{(-1)}^{\left[x\right]}$

Replace $x\to 0-h\text{or}x\to -h,\text{where}h>0$

$\underset{h\to 0}{\lim }{(-1)}^{[-h]}$

Applying the limits,

$$\begin{gathered} \underset{h\to 0}{\lim }{(-1)}^{[-h]} \\ ={(-1)}^{-1} \\ =-1 \end{gathered}$$

Step 2: Expanding the given equation and applying the limits

Now, replace $x\to 0+h\text{or}x\to h,\text{where}h>0$

$\underset{h\to 0}{\lim }{(-1)}^{[h+0]}$

Applying the limits,

$$\begin{gathered} \underset{h\to 0}{\lim }{(-1)}^{[h+0]} \\ ={(-1)}^0 \\ =1 \end{gathered}$$

As, $\underset{h\to 0}{\lim }{(-1)}^{[-h]}\ne \underset{h\to 0}{\lim }{(-1)}^{[h+0]}$, the limits do not exist.

Therefore, option (D) is the correct answer.