Animated Solution for Mathematics - Limits, Continuity and Differentiability: ${\lim }_{x\to 0}\frac{\log x^n-[x]}{[x]},n\in N$, ($[x]$ denotes greatest integer less than or equal to $x$)

We are tasked with evaluating the limit:

where $n\in \mathbb{N}$. At first glance, one might be tempted to apply standard tools like L'Hopital's Rule or Taylor expansions. However, the presence of the Greatest Integer Function, $[x]$, serves as a critical warning sign.

This function is not smooth; it represents a staircase of discontinuities. We must analyze the behavior of the expression by examining the left-hand and right-hand limits separately.

Let us investigate the approach from the right, $x\to 0^{+}$. In the interval $0<x<1$, the definition of the Greatest Integer Function dictates that $[x]=0$.

Substituting this into our expression, we obtain:

We are not dealing with a limit that approaches zero; we are dealing with a denominator that is identically zero. In the realm of real numbers, division by zero is undefined. Consequently, the Right-Hand Limit (RHL) does not exist.

Next, we consider the approach from the left, $x\to 0^{-}$. In the interval $(-1,0)$, the greatest integer less than or equal to $x$ is $[x]=-1$.

Substituting this value into the expression, we get:

The behavior of this term depends entirely on the parity of $n$:

1. If $n$ is odd, $x^n$ is negative for $x<0$. Since the logarithm of a negative number is undefined in the real number system, the limit does not exist. 2. If $n$ is even, $x^n$ is positive. As $x\to 0^{-}$, $x^n$ approaches $0^{+}$, which forces $\log (x^n)$ to approach $-\infty$. Thus, the expression approaches $+\infty$.

For a limit to exist, the RHL and the LHL must both exist and be equal. We have demonstrated that the RHL is undefined due to division by zero, and the LHL is either undefined or infinite.

There is no common ground for these values. Therefore, the function is fundamentally discontinuous and undefined at $x=0$.