91Ó°ÊÓ

When exploring the concept of limits, one-sided limits provide insight into the behavior of functions as they approach a specific point from one direction only. A limit from the right (denoted as \(x \to a^+\)) considers values of \(x\) slightly greater than \(a\), while a limit from the left (\(x \to a^-\)) looks at values slightly smaller.

Calculating both one-sided limits helps determine if a two-sided limit exists. For instance, if \(\lim_{x \to 1^+} f(x) = L_1\) and \(\lim_{x \to 1^-} f(x) = L_2\), the two-sided limit \(\lim_{x \to 1} f(x)\) only exists if \(L_1 = L_2\).

This principle is crucial in the exercise, where the right limit is \(-\frac{1}{2}\) and the left limit is \(\frac{1}{2}\), showing distinct behaviors from both sides, leading to non-existence of the two-sided limit.

The absolute value function, denoted by \(|x|\), is essential in calculus for its non-negative nature. It represents the distance of a number from zero on the number line without regard to direction.

The function is defined as:

• \(|x| = x\) if \(x \geq 0\)
• \(|x| = -x\) if \(x < 0\)

This property of flipping a negative sign to positive affects how limits behave, as seen in the given exercise.

For \(x \to 1^+\), \(|x-1| = x-1\) because values are slightly greater than 1. For \(x \to 1^-\), \(|x-1| = -(x-1)\) because values are slightly less than 1. This distinction is crucial in calculating one-sided limits and understanding discontinuities.

A quadratic expression takes the form \(ax^2 + bx + c\). In our exercise, the quadratic term \(1-x^2\) can be expanded to \(-(x-1)(x+1)\). Recognizing its structure is helpful for limit evaluation.

Quadratics have a definitive parabolic shape, and in this case, \(1-x^2\) leads to a downward opening parabola intersecting the x-axis at \(x = 1\) and \(x = -1\). These roots are identified by solving \(1-x^2 = 0\) or factoring as seen.

Understanding these roots assists in identifying critical points of behavior change in functions and aids in the examination of discontinuities.

Discontinuity refers to points where a function is not continuous. A very useful concept in calculus, discontinuities can occur at points, over intervals, or even be infinite.

In our exercise, the function \(\frac{|x-1|}{1-x^2}\) exhibits a point discontinuity at \(x = 1\). This is evident as the one-sided limits from the left and right do not agree: the right-side limit is \(-\frac{1}{2}\) and the left-side limit is \(\frac{1}{2}\). Such a discontinuity indicates the function's value cannot be continuously defined at that point.

Understanding this helps identify where a function changes its behavior dramatically, which is crucial in both theoretical and applied calculus.