Question

, find each of the right-hand and left-hand limits or state that they do not exist. $$ \lim _{x \rightarrow 0^{-}} \frac{x}{|x|} $$

Short answer

The left-hand limit is -1.

Step-by-step solution

Identify the Direction of the Limit

The expression requires evaluating the limit as \( x \) approaches 0 from the left, denoted by \( x \to 0^- \). This means we consider values of \( x \) that are small and negative, such as -0.1, -0.01, etc.

Simplify the Expression for x Approaching 0 from the Left

For \( x < 0 \), the absolute value \( |x| \) is equal to \( -x \). Thus the expression \( \frac{x}{|x|} \) becomes \( \frac{x}{-x} = -1 \) when \( x \) is negative.

Evaluate the Limit

Since the simplified expression \( \frac{x}{|x|} \) equals \( -1 \) for any \( x < 0 \), the limit as \( x \to 0^- \) is \( -1 \). This is because the function value does not change as \( x \) approaches 0 from the negative side.

Key concepts

Left-Hand Limit

The left-hand limit is a fundamental concept in calculus, particularly useful when addressing the behavior of functions as they approach specific points from the left side on a number line. A left-hand limit is denoted by \( \lim_{x \to c^{-}} f(x) \), where \( x \) approaches a number \( c \) from values less than \( c \). This is why it is often called "approaching from the left."

Let's consider the expression \( \lim_{x \to 0^-} \frac{x}{|x|} \). Here, \( x \) approaches 0 from the negative side, meaning values close to 0 but less than 0, like -0.1 or -0.01. For such negative values, the absolute value \( |x| \) is \( -x \), leading to the expression \( \frac{x}{-x} = -1 \).

So, the left-hand limit \( \lim_{x \to 0^-} \frac{x}{|x|} = -1 \). This means as we approach 0 from the left, the function consistently hauls towards -1, indicating a stable value on that side.

Directional Limits

Directional limits provide a more detailed view of limits by focusing on the behavior of functions as they come from only one side of a given point, enhancing our understanding beyond simply evaluating overall limits. They are particularly useful when functions have points of discontinuity or when they behave differently from the left and the right. We use this approach to assure that the function's behavior, from both directions, aligns or contrasts.

For example, evaluating a function like \( f(x) = \frac{x}{|x|} \) at \( x = 0 \) could appear challenging due to potential discontinuity precisely at zero. In such cases, we analyze limits from both directions (left-hand and right-hand). The left-hand limit \( \lim_{x \to 0^-} \) results in -1, as explored earlier. However, the right-hand limit \( \lim_{x \to 0^+} \) would need a separate evaluation to assess whether it aligns with the left-hand limit, providing clarity about continuity and defining the full linear behavior at zero.

Mathematical Proofs

In calculus, mathematical proofs serve not only as validation of mathematical statements but are also educational exercises in logical thinking and problem-solving. They help in establishing the truth of concepts such as limits, continuity, derivatives, and integrals. Proofs often involve an understanding of limit properties and careful examination of values that approach a function from various directions.

When proving directional limits, as with the exercise \( \lim_{x \to 0^-} \frac{x}{|x|} = -1 \), one ensures that for negative values approaching 0, the relationship consistently holds. We simplify \( \frac{x}{|x|} \) with \( x < 0 \), showing the expression unequivocally results in -1 for all such \( x \).

Detailed mathematical proofs not only reinforce why a limit results in a specific value but also highlight the conditions under which it holds true and the mathematical reasoning that supports such conclusions. This rigorous approach solidifies understanding and cements the concepts, ensuring robust mathematical comprehension.