To find the limit of the expression \( x \ln \left( \frac{1}{x} \right) \) as \( x \) approaches \( 0^+ \), we can rewrite the expression and analyze it step-by-step.

Given:
\[ \lim_{x \to 0^+} x \ln \left( \frac{1}{x} \right) \]

First, let's rewrite the logarithmic term:
\[ \ln \left( \frac{1}{x} \right) = \ln (1) - \ln (x) = -\ln (x) \]

So the expression becomes:
\[ \lim_{x \to 0^+} x (-\ln (x)) = - \lim_{x \to 0^+} x \ln (x) \]

Now we focus on finding the limit of \( x \ln (x) \) as \( x \) approaches \( 0^+ \).

We can use substitution to simplify the problem. Let:
\[ t = \ln (x) \]
As \( x \) approaches \( 0^+ \), \( t \) approaches \( -\infty \). Note that \( x = e^t \).

Substitute \( x \) with \( e^t \):
\[ x \ln (x) = e^t \cdot t \]

Now consider the limit:
\[ \lim_{t \to -\infty} e^t \cdot t \]

As \( t \to -\infty \), \( e^t \to 0 \) and \( t \to -\infty \). However, the exponential function \( e^t \) approaches zero much faster than \( t \) approaches \( -\infty \). Therefore, \( e^t \cdot t \) approaches zero.

Hence,
\[ \lim_{x \to 0^+} x \ln (x) = 0 \]

Thus,
\[ \lim_{x \to 0^+} x \ln \left( \frac{1}{x} \right) = 0 \]

If you have any questions or need further details, feel free to ask!

Here are some related questions for further practice:
1. What is the limit of \( x \ln (x) \) as \( x \) approaches infinity?
2. How does L'Hôpital's Rule apply to limits involving \( \ln (x) \)?
3. What is the limit of \( x e^x \) as \( x \) approaches 0?
4. What are the properties of the natural logarithm function that help in limit calculations?
5. How do you find the limit of \( x^2 \ln (x) \) as \( x \) approaches 0 from the right?
6. What is the limit of \( x \ln (x^2) \) as \( x \) approaches 0 from the right?
7. How does substitution help in evaluating complex limits?
8. What is the limit of \( \frac{\ln (x)}{x} \) as \( x \) approaches infinity?

**Tip:** When evaluating limits involving logarithms, consider substitutions that simplify the expressions, such as changing variables to transform the logarithmic terms into linear or exponential forms.

Explore the limit of the expression x ln(1/x) as x approaches 0 from the positive side. Understand the step-by-step solution involving logarithms and exponential functions. Suitable for college-level students and anyone interested in calculus and limit problems.

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