1. Interpret the expression carefully

We need to evaluate

where $[x]$ is the greatest integer function.

For real logarithm, $\log(x^n)$ is defined near $x=0$ only when $x^n>0$.

• If $n$ is even, then for $x\neq 0$, $x^n>0$, so both sides are allowed.
• If $n$ is odd, then for real log, we need $x>0$, so only the right-hand side is relevant.

In either case, to test existence of the limit, it is enough to inspect values of $[x]$ near $0$.

2. Behavior of $[x]$ near $x=0$

For $x\to 0^+$), we have $0<x<1$, hence

Then the given expression becomes

which is not defined. So there are no domain points arbitrarily close to $0$ from the right where the expression is defined.

For $x\to 0^-$, we have $-1<x<0$, hence

Then

Now:

• if $n$ is even, $x^n>0$ and as $x\to 0^-$, $$\log(x^n)=n\log|x|\to -\infty,$$ so $$-\log(x^n)-1\to +\infty.$$
• if $n$ is odd, $\log(x^n)$ is not real-defined for $x<0$.

So in no case do we get a finite two-sided limit.

3. Conclusion

The expression is not defined on the right side near $0$ because $[x]=0$ for $0<x<1$, making the denominator zero. Hence the required limit does not exist.

Therefore, the correct option is

4. Comparison with stored answer

Stored correct answer: D.

My derived answer is also D, so they agree.