Problem 36 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(31 - 36 ,\) determine the limit. $$\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { | x | }$$

Step-by-Step Solution

Answer

The limit of \( \frac { x } { | x | } \) as \( x \) approaches 0 from the left is -1.

Step 1: Understand the absolute value function

First, understand that for \( x < 0 \) , \( | x | = - x \). This is because the absolute value of a number is its distance from 0 on the number line irrespective of direction. When \( x \) is less than 0, the absolute value is the positive counterpart of \( x \). So, we can rewrite the function as \(\frac { x } { -x } \) for \( x < 0 \).

Step 2: Simplify the function

Next, simplify the function \(\frac { x } { -x } \). Both numerator and denominator will cancel out leaving -1.

Step 3: Find the limit of the function

Now we can find the limit of the function as \( x \) approaches 0 from the left side. The function is constant and equal to -1 for \( x < 0 \). Hence, it does not matter what value \( x \) is approaching from the left side, the function will always yield -1. Thus, the limit of the function as \( x \) approaches 0 from the left side is -1.

Key Concepts

Absolute Value

The concept of absolute value is crucial for understanding functions like the one in this exercise. Absolute value is essentially the distance of a number from zero on the number line, irrespective of its direction. This means:

• If a number is positive, its absolute value is the number itself.
• If a number is negative, its absolute value is the positive version of that number.

For example, the absolute value of -3 is 3, because the distance from zero is three units, regardless of direction.
The function given in the exercise involves the expression \(|x|\). When dealing with this kind of expression, it's important to split into cases depending on whether \(x\) is positive, negative, or zero. For negative values, like in this exercise where \(x < 0\), \(|x|\) is equal to \(-x\). This transformation is essential to simplify expressions and solve limits involving absolute values.

One-Sided Limits

One-sided limits help us understand the behavior of a function as it approaches a specific point from one direction, either from the left or the right. These are different from regular limits, which consider both directions simultaneously.
In this case, we are dealing with the left-hand limit symbolized like this: \( \lim _ { x \rightarrow 0 ^ { - } } \). This notation tells us that \(x\) is approaching 0 from values less than 0, i.e., from the negative side.
Understanding one-sided limits is particularly useful when a function behaves differently approaching a point depending on the direction, or in scenarios where a function may not even exist from one side. The one-sided limit in our exercise shows that even if \(x\) approaches 0 from the negative side, the function approaches a constant value of -1.

Calculus Limits

The concept of limits in calculus represents the value that a function approaches as its input approaches a certain point. Limits are foundational for defining other concepts like continuity, derivatives, and integrals.
In some cases, functions may not reach a particular value at points of interest but will approach it as closely as desired. Calculating limits involves:

• Evaluating the function at particular approaches, like approaching a number from the right or left.
• Simplifying expressions to determine what value the function is reaching.

In our problem, the limit being sought is when \(x\) tends to zero from the left-hand side (\(0^-\)). By simplifying \(\frac{x}{|x|}\), we find that the function is constantly -1 for all negative values of \(x\), leading to the conclusion that the left-hand limit as \(x\) approaches zero is -1.
Limits are powerful in handling calculations where straightforward evaluation is not possible, such as when the function is undefined or indeterminate at a point.