Limit x tends to infinity (logx^n - [x])/[x] is.Here, [.] Is a step function

To evaluate the limit of the expression \((\log(x^n) - [x])/[x]\) as \(x\) approaches infinity, where \([x]\) denotes the greatest integer less than or equal to \(x\), we can break it down step by step.

Understanding the Components

First, let's clarify the components of the expression:

• \(\log(x^n)\): This can be simplified using the properties of logarithms. Specifically, \(\log(x^n) = n \log(x)\).
• \([x]\): This is the greatest integer function, which means it rounds down \(x\) to the nearest integer. As \(x\) approaches infinity, \([x]\) will be very close to \(x\), specifically \([x] = x - \{x\}\), where \(\{x\}\) is the fractional part of \(x\).

Rewriting the Expression

Now, we can rewrite the limit expression:

\[ \lim_{x \to \infty} \frac{n \log(x) - [x]}{x} \]

Substituting \([x]\) with \(x - \{x\}\), we get:

\[ \lim_{x \to \infty} \frac{n \log(x) - (x - \{x\})}{x} = \lim_{x \to \infty} \frac{n \log(x) - x + \{x\}}{x} \]

Breaking Down the Limit

This can be further simplified to:

\[ \lim_{x \to \infty} \left(\frac{n \log(x)}{x} - 1 + \frac{\{x\}}{x}\right) \]

Evaluating Each Term

Now, let's evaluate each term in the limit:

• Term 1: \(\frac{n \log(x)}{x}\)

As \(x\) approaches infinity, \(\log(x)\) grows much slower than \(x\). Thus, \(\frac{n \log(x)}{x} \to 0\).

• Term 2: \(-1\)

This term remains constant as we take the limit.

• Term 3: \(\frac{\{x\}}{x}\)

The fractional part \(\{x\}\) is always less than 1, so \(\frac{\{x\}}{x} \to 0\) as \(x\) approaches infinity.

Combining the Results

Putting it all together, we have:

\[ \lim_{x \to \infty} \left(0 - 1 + 0\right) = -1 \]

Final Result

Therefore, the limit of the given expression as \(x\) tends to infinity is:

-1