Problem 41 - Brief Applied Calculus

Question

\(41-44\). For each function, find: a. \(\lim _{x \rightarrow 0^{-}} f(x)\) b. \(\lim _{x \rightarrow 0^{+}} f(x)\) c. \(\lim _{x \rightarrow 0} f(x)\) $$ f(x)=|x| $$

Step-by-Step Solution

Answer

a. 0 b. 0 c. 0

Step 1: Understanding the Function

First, understand that the function given is the absolute value function \( f(x) = |x| \). This function returns the non-negative value of \( x \).

Step 1: Finding \( \lim_{x \to 0^-} f(x) \)

For \( \lim_{x \to 0^-} f(x) \), we consider values of \( x \) that are approaching 0 from the negative side. Since the function \( f(x) = |x| \) returns the positive value, \( |x| = -x \) for \( x < 0 \). Thus, \( \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x) = 0 \).

Step 2: Finding \( \lim_{x \to 0^+} f(x) \)

For \( \lim_{x \to 0^+} f(x) \), consider values of \( x \) that are approaching 0 from the positive side. In this case, \( f(x) = |x| = x \) for \( x > 0 \). Therefore, \( \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x = 0 \).

Step 3: Finding \( \lim_{x \to 0} f(x) \)

The limit \( \lim_{x \to 0} f(x) \) exists if both the left-hand limit \( \lim_{x \to 0^-} f(x) \) and the right-hand limit \( \lim_{x \to 0^+} f(x) \) are equal. Since both limits are 0, the limit \( \lim_{x \to 0} f(x) = 0 \).

Key Concepts

Understanding the Absolute Value Function

The absolute value function, denoted as \( |x| \), is quite straightforward. It takes any real number \( x \) and returns the distance of \( x \) from zero on the number line. This means that the absolute value is always non-negative.
For example, both \( |3| \) and \( |-3| \) are equal to 3. If \( x \) is positive or zero, \( |x| \) remains \( x \). However, if \( x \) is negative, \( |x| \) becomes \( -x \), because the function outputs the positive equivalent.
This nature of the absolute value function makes it crucial in calculus, especially when considering limits involving values approaching zero from different sides.

Exploring One-Sided Limits

One-sided limits help us understand the behavior of functions as the input approaches a particular point from one direction only.

• The left-hand limit or \( \lim_{x \to c^-} f(x) \) examines values of \( x \) approaching \( c \) from the left (or negative) side. For \( f(x) = |x| \), approaching zero from the left means considering \( x \) values just below zero, like \(-0.1\).
• The right-hand limit or \( \lim_{x \to c^+} f(x) \) checks values of \( x \) coming from the right (or positive) side. For \( f(x) = |x| \), this involves values slightly greater than zero, such as \(0.1\).

For \( |x| \), both approaches yield the same result for \( c = 0 \), highlighting the symmetry about the y-axis and the function’s ability to convert negative values to positive.

Evaluating Limits Involving Absolute Values

Evaluating limits, particularly when absolute values are involved, asks the question: "What value does the function approach as \( x \) gets infinitely close to a specific point from both sides?"
To determine the overall limit \( \lim_{x \to c} f(x) \), where \( c \) is our point of interest, we consider:

• \( \lim_{x \to c^-} f(x) \) — This checks the function's behavior as \( x \) approaches \( c \) from the left. For \( f(x) = |x| \), \( \lim_{x \to 0^-} f(x) = 0 \), since \( |x| \) behaves as \( -x \) and approaches zero from the negative side.
• \( \lim_{x \to c^+} f(x) \) — This investigates the function's path coming from the right. Here, \( \lim_{x \to 0^+} f(x) = 0 \), because \( |x| = x \) when \( x \) gets small from the positive side.
• Overall Limit: The two one-sided limits are compared, and an overall limit exists only when these two are equal, ensuring continuity at that point.

Because both left-hand and right-hand limits reach the same result of 0 for this scenario, \( \lim_{x \to 0} |x| = 0 \), confirming that \(|x|\) is continuous at zero.