91Ó°ÊÓ

When tackling calculus problems involving limits, one-sided limits are crucial. These limits focus on approaching a particular point from just one direction (right-hand or left-hand). In our exercise, we deal with the limit as \( x \to 1^- \), which means we are interested in how the function behaves as \( x \) approaches 1 from the left side. One-sided limits help us understand the behavior of functions at points of discontinuity or where the function is undefined when approaching from one direction.

To determine a one-sided limit, we must:

• Consider only values of \( x \) that are slightly less than or greater than the point of interest.
• Evaluate the behavior of the function as \( x \) moves towards this point.

This perspective is essential when dealing with functions that might approach very large positive or negative values, or where a two-sided limit might not exist.

The natural logarithm, denoted as \( \ln(x) \), is a logarithmic function with the base \( e \). The constant \( e \) is approximately equal to 2.71828. This function is fundamental in mathematics due to its natural growth properties. For values of \( x \) between 0 and 1, \( \ln(x) \) is negative since the base \( e \) raised to any power between 0 and 1 results in a fraction.

The natural logarithm has the following important properties:

• \( \ln(1) = 0 \)
• \( \ln(x) > 0 \) for any \( x > 1 \)
• \( \ln(x) < 0 \) for any \( 0 < x < 1 \)

In our specific case, as \( x \to 1^- \), \( \ln(x) \) nears 0 from the negative side. Since the natural logarithm can only operate on positive numbers, understanding its behavior just before 1 helps us deduce the behavior of compositions, like \( \ln(\ln(x)) \).

Infinite limits arise when functions grow without bound. They don't settle into a finite number but instead approach \( \infty \) or \(-\infty \). In our exercise, as \( x \to 1^- \), we observe that \( \ln(\ln(x)) \) tends toward \(-\infty \).

Here's how this occurs:

• As \( x \to 1^- \), \( \ln(x) \to 0^- \), meaning slightly less than 0.
• Then, when we apply the outer logarithm function \( \ln(\ln(x)) \), it reacts to this negative value (\( u \to 0^- \)) by approaching \(-\infty \).

This concept highlights scenarios in calculus where values escape to infinity in certain directions, prompting us to conclude behavior like the limit \( \lim_{x \to 1^-} \ln(\ln(x)) = -\infty \). Understanding infinite limits provides deeper insight into how functions can behave very differently depending on the path of approach.