91Ó°ÊÓ

The natural logarithm, denoted as \( \ln \), is a fundamental concept in calculus and mathematics. It is the logarithmic function with the base \( e \), where \( e \) is approximately 2.71828—a constant known as Euler's number. This function is the inverse of the exponential function, meaning if you have \( e^y = x \), then \( \ln x = y \).
Understanding the properties of the natural logarithm helps in evaluating functional expressions like \( x \ln x \). Some important properties include:

• \( \ln 1 = 0 \)
• \( \ln(ab) = \ln a + \ln b \)
• \( \ln(a^b) = b \ln a \)

For small values of \( x \), as we approach zero from the positive side, \( \ln x \) becomes increasingly negative. This is an important characteristic to consider when evaluating limits that involve natural logarithms, particularly in expressions going towards a limit.

Function evaluation is the process of substituting a specific value of \( x \) into a function to find the corresponding \( y \)-value.
In our exercise, we substitute small positive values like 0.1, 0.01, and so on into the function \( f(x) = x \ln x \).
This allows us to observe how the function behaves as \( x \) approaches zero from the positive side.
Here is a step by step breakdown of how this evaluation works:

• Substitute \( x \) into \( \ln x \) using a calculator or tables to find \( \ln(0.1), \ln(0.01), \ldots \)
• Multiply the result by \( x \) yielding \( f(x) = x \ln x \)
• Repeat the process for each value needed to observe the behavior of \( f(x) \)

By evaluating the function at these small values, we calculate the output which assists in estimating the limit. Noticing the pattern of \( f(x) \) becoming closer to zero provides insights into the solution's direction and the answer for the limit.

Limit estimation is a fundamental technique in calculus used to determine the behavior of a function as the input approaches a particular point, in this case, as \( x \to 0^+ \). In this process, we observe trends in function outputs as inputs get closer to a specified value.
Our goal here is to determine \( \lim_{x \to 0^+} x \ln x \). As seen in the function evaluations, as \( x \) decreases toward zero, the product \( x \ln x \) gets smaller and approaches zero. The steps to estimate limits are:

• Perform function evaluations at increasingly smaller values of \( x \)
• Observe the pattern or trend in function outputs
• Consider how the function behavior aligns with mathematical concepts, such as infinity and convergence

In this exercise, the repeated evaluations show a consistent trend of \( f(x) \) values converging towards zero. This convergence heuristic helps in hypothesizing that the limit of \( x \ln x \) as \( x \to 0^+ \) equals zero, giving us a solid conclusion.