The question is to evaluate the limit, where [•] denotes the greatest integer function.

Question

$$\lim_{x\to 0} \frac{(\log_e x^n)-\lfloor x\rfloor}{x}$$

I tried separating the two functions in the numerator but then I get stuck up with the greatest integer part.

Answer 1

The function $y=x^n$ isn't defined for $x<0$.

Thus, we'll only discuss the case when $x>0$.

Since greatest integer of any number $\in (0,1)=0$. Hence, $\lfloor x \rfloor$, when $x\to 0^+$ is equal to $0$.

Thus \begin{align} \lim_{x\to 0} \frac{(\log_e x^n)-\lfloor x\rfloor}{x} &=\lim_{x\to 0} \frac{(\log_e x^n)}{x}\\ &=\lim_{x\to 0} \frac{n (\log_e x)}{x}\\ &=n \cdot \lim_{x\to 0} \frac{(\log_e x)}{x}\to n \cdot -\infty \to -\infty \end{align}

Thus, the limit doesn't exist.