91Ó°ÊÓ

One-sided limits are an essential concept when evaluating limits in calculus. They help us understand the behavior of a function as the variable approaches a specific point, either from the left or right. This is especially useful when dealing with functions that may have different behaviors on either side of a point. In our problem, we are evaluating the limit of the function \( f(x) = \frac{x}{|x|} \) as \( x \) approaches 0.

To understand one-sided limits better, consider:

• Left-Hand Limit (\( \lim_{x \to c^-} f(x) \)): This represents the value that the function \( f(x) \) approaches as \( x \) comes towards \( c \) from values smaller than \( c \). In our example, this is \( x \to 0^- \).
• Right-Hand Limit (\( \lim_{x \to c^+} f(x) \)): This represents the value \( f(x) \) approaches as \( x \) comes towards \( c \) from values greater than \( c \). For our function, this is \( x \to 0^+ \).

By calculating these limits, we can determine if the overall limit exists. If the left-hand and right-hand limits are equal, the two-sided limit exists. If not, the overall limit does not exist. In our problem, since the left and right limits are different, the overall limit at \( x = 0 \) does not exist.

The absolute value function is crucial in calculus and many other mathematical concepts. It is denoted by \( |x| \), representing the distance of \( x \) from zero on the number line, without considering direction. Essentially, the absolute value of a number is the number itself if it is positive, or the negation of the number if it is negative.

In expression form: - If \( x \) is positive or zero: \( |x| = x \)- If \( x \) is negative: \( |x| = -x \)
This function introduces a piecewise component where different expressions define the function depending on the sign of \( x \). In our exercise, the function \( f(x) = \frac{x}{|x|} \) becomes different rational expressions for both positive and negative values of \( x \).

Understanding how \( |x| \) operates allows us to analyze how the function behaves differently as \( x \) approaches 0 from either side, leading to different one-sided limits.

Discontinuity in a function occurs at points where the function is not continuous, meaning there is a break, gap, or jump in the function's graph. Understanding discontinuity involves identifying places where the function's limit does not match the function's value or is not defined.

The expression \( f(x) = \frac{x}{|x|} \) shows a discontinuity at \( x = 0 \). Here's why:

• As \( x \to 0^- \), the function approaches -1.
• As \( x \to 0^+ \), the function approaches 1.

These different approach values indicate a jump discontinuity, where the function "jumps" from one value (-1) to another (1) without being smooth or defined at \( x = 0 \) itself.

Discontinuity points such as these emphasize the importance of one-sided limits, allowing us to analyze separately from either direction to understand the precise nature of a point where a function isn't straightforward.