91Ó°ÊÓ

Infinite limits occur when a function approaches an infinitely large positive or negative value as the input gets closer to a particular point. In the context of the exercise, we're examining what happens to the function \( f(x) = 2\ln|x| - \frac{1}{x^2} \) as \( x \) approaches zero. This involves considering the behavior of the natural logarithm and the reciprocal squared function.When approaching zero, the term \( \frac{1}{x^2} \) zooms towards infinity since the denominator becomes exceedingly small. Consequently, \( -\frac{1}{x^2} \) tends to negative infinity. Meanwhile, \( 2\ln|x| \) also heads towards negative infinity as \( x \) nears zero. Combined, these behaviors mean that the overall function races towards \(-\infty\), illustrating an infinite limit.

The natural logarithm, denoted as \( \ln(x) \), is a special type of logarithm that uses the base \( e \), approximately equal to 2.718. The function is only defined for positive values of \( x \), and it is crucial for understanding certain limits, including infinite limits, as shown in the exercise.Properties of Natural Logarithm:

• \( \ln(1) = 0 \): The value of ln at 1 is always zero.
• \( \ln(x)\) becomes negative as \( x \) approaches zero from the positive side.
• \( \ln(x) \) tends towards infinity as \( x \) increases indefinitely.

In the problem, \( 2\ln|x| \), with \( |x|^2 = x^2 \), tends towards \(-\infty\) as \( x \to 0 \). This is because the square of a small number approaches zero, and thus its logarithm approaches negative infinity.

Limit analysis is the method of evaluating what value a function approaches as the input nears a particular point. In this exercise, the goal is to determine the infinite limit as \( x \to 0 \) for the function \( f(x) = \ln(x^2) - x^{-2} \).There are several key steps in limit analysis:

• Separate Analysis: Break down the function into its components; here it's \( \ln(x^2) \) and \( -x^{-2} \).
• Behavior Prediction: Predict how each component behaves individually as \( x \to 0 \).
• Combine Results: Evaluate the combination of these predictions to determine the overall behavior of the function.

By following these steps, as \( x \to 0 \), we identified that both \( 2\ln|x| \) and \( -\frac{1}{x^2} \) spiral down to \(-\infty\), culminating in the limit of \( -\infty \). This thorough analysis simplifies understanding and verification of such calculus problems.