Question

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 0^{-}} \frac{|x|}{x} $$

Short answer

-1

Step-by-step solution

Recognize the Absolute Value

The absolute value of \(x\) (denoted \(|x|\)) is defined as \(x\) if \(x > 0\) or \(-x\) if \(x < 0\).

Express the Absolute Value Properly

Considering that \(x\) approaches 0 from the left (indicated by \(0^-\)), we can say that \(x\) will be slightly less than 0. Therefore, the absolute value of \(x\), i.e., \(|x|\), can be expressed as \(-x\) in this case.

Substitute the Absolute Value

After substituting \(-x\) for \(|x|\), the original limit expression \(\lim _{x \rightarrow 0^{-}} \frac{|x|}{x}\) becomes \(\lim _{x \rightarrow 0^{-}} \frac{-x}{x}\).

Simplify and Evaluate the Limit

Upon simplifying, the equation \(\lim _{x \rightarrow 0^{-}} \frac{-x}{x}\) reduces to \(\lim _{x \rightarrow 0^{-}} -1\), which simply equals -1, as a constant's limit is the constant itself.

Key concepts

Absolute Value

The absolute value of a number measures its distance from zero on the number line, without considering which direction (positive or negative) it lies. It's a helpful concept because it allows us to deal with values that could be positive or negative while focusing on their magnitude.

For any real number, denoted as \( x \), the absolute value is expressed as \( |x| \). This can be defined in two cases:

• If \( x > 0 \), then \( |x| = x \)
• If \( x < 0 \), then \( |x| = -x \)

Thus, the absolute value transforms any negative input into a positive output, mirroring negative numbers across the origin.

Limit Laws

The study of limits is foundational in calculus, as it underpins both derivatives and integrals. Limit laws provide us with a set of guidelines to compute limits more conveniently. They are akin to rules of arithmetic for limits.

Some basic limit laws include:

• Constant Law: The limit of a constant is just the constant itself.
• Sum/Difference Law: The limit of a sum/difference of functions is the sum/difference of their limits.
• Product Law: The limit of a product is the product of the limits.
• Quotient Law: The limit of a quotient is the quotient of the limits, provided the limit of the denominator isn't zero.

By using these laws, solving limit problems becomes a systematic process, enabling us to break down complex expressions into simpler components that we can handle individually.

One-Sided Limits

One-sided limits focus on the behavior of a function as it approaches a specified point from only one direction (either from the left or from the right). These are vital when determining the behavior of functions near points where they might not be defined on both sides.

For example, when calculating \( \lim_{x \to 0^-} \frac{|x|}{x} \), we look at the value \( x \) assumes as it approaches zero from the left, which means through negative values. Determining one-sided limits helps us understand sudden changes or jumps in function values that could indicate discontinuities or other interesting features.

Piecewise Functions

Piecewise functions are composed of different sub-functions, each applying to specific intervals of the main function's domain. These functions allow for modeling scenarios where behavior changes across different conditions.

Consider the absolute value function as a simple form of piecewise function:

• When \( x > 0 \), the function simplifies to \( f(x) = x \).
• When \( x < 0 \), it is defined as \( f(x) = -x \).

An understanding of piecewise functions is crucial, especially when dealing with limits, as it requires determining which piece of the function applies at the point of interest. Handling them often involves analyzing each sub-function separately and connecting their behaviors at each interval’s boundaries.