91Ó°ÊÓ

When we talk about limits in calculus, one-sided limits allow us to focus on the behavior of functions as they approach a specific value from just one side. This concept is particularly useful when analyzing a point where a function might behave differently from the left and right.

As seen in our problem with the function \( f(x) = -|x| \), understanding one-sided limits is critical. To find the limit as \( x \) approaches 0 from the left, represented as \( \lim_{x \to 0^{-}} \), we check how our function behaves when we use values slightly less than zero. For \( f(x) = -|x| \), \(|x|\) is equal to \(-x\) for negative \( x \), making \( f(x) = x \). Thus, as \( x \to 0^{-} \), \( f(x) \to 0 \).

On the other hand, finding the limit as \( x \) approaches 0 from the right, \( \lim_{x \to 0^{+}} \), involves values slightly greater than zero. Here, \( |x| \) remains \( x \), meaning \( f(x) = -x \). Again, as \( x \to 0^{+} \), \( f(x) \to 0 \). These separate analyses help confirm that the one-sided limits are equal.

The absolute value function is quite fascinating. It is designed to give the non-negative value of any real number. Mathematically, it's represented as \( |x| \), and its interpretation changes depending on whether \( x \) is positive or negative.

Consider our function \( f(x) = -|x| \). Here, the absolute value component makes the function output the negative of whatever absolute value it encounters. So, for positive \( x \), \( |x| = x \) and thus \( f(x) = -x \). For negative \( x \), \( |x| = -x \) hence \( f(x) = x \).

The absolute value function has a characteristic V-shape when graphed, which makes the analysis of limits at any point more intuitive visually. Our function \( -|x| \) has an inverted V shape. This understanding simplifies working out one-sided limits or any abrupt changes in the function as it approaches specific points.

Continuity is an important concept in calculus, detailing how a function behaves at any given point. A function is continuous at a point if three conditions meet:

• The function is defined at the point.
• The limit of the function as it approaches the point exists.
• The limit equals the function's value at that point.

Let's check our function \( f(x) = -|x| \) for continuity at \( x = 0 \). We've found:

- \( \lim_{x \to 0^{-}} f(x) = 0 \)
- \( \lim_{x \to 0^{+}} f(x) = 0 \)

Both one-sided limits are equal, so we also have \( \lim_{x \to 0} f(x) = 0 \). Since \( f(0) = 0 \), all conditions for continuity are satisfied. Hence, \( f(x) \) is continuous at \( x = 0 \), creating a smooth transition with no jumps or gaps at this point. This allows us to trust that the function behaves predictably at \( x = 0 \) just as it does elsewhere in its domain.