Problem 42 - Brief Applied Calculus

Question

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

All limits are 0: \(\lim _{x \rightarrow 0^{-}} f(x) = \lim _{x \rightarrow 0^{+}} f(x) = \lim _{x \rightarrow 0} f(x) = 0\).

Step 1: Identify the Function Components

The function given is \(f(x) = -|x|\). The absolute value function \(|x|\) is defined as \( |x| = x \) when \( x \geq 0 \) and \( |x| = -x \) when \( x < 0 \). Therefore, \(-|x|\) becomes \(-x\) when \(x \geq 0\) and \(x\) when \(x < 0\).

Step 2: Evaluate \(\lim _{x \rightarrow 0^{-}} f(x)\)

For \(x\) approaching 0 from the left \((x \rightarrow 0^-)\), \(f(x) = x\). Thus, \[\lim_{x \rightarrow 0^-} f(x) = \lim_{x \rightarrow 0^-} x = 0.\]

Step 3: Evaluate \(\lim _{x \rightarrow 0^{+}} f(x)\)

For \(x\) approaching 0 from the right \((x \rightarrow 0^+)\), \(f(x) = -x\). Thus,\[\lim_{x \rightarrow 0^+} f(x) = \lim_{x \rightarrow 0^+} (-x) = 0.\]

Step 4: Evaluate \(\lim _{x \rightarrow 0} f(x)\)

Since both the left-hand limit \(\lim_{x \rightarrow 0^-} f(x)\) and the right-hand limit \(\lim_{x \rightarrow 0^+} f(x)\) are equal and both are 0, we conclude:\[\lim_{x \rightarrow 0} f(x) = 0.\]

Key Concepts

Left-Hand Limit

A left-hand limit refers to approaching a specific point from values less than the point itself, primarily seen as approaching the point from the left side on the graph. In calculus, this is expressed as \( \lim_{x \to c^-} f(x) \). Here, we examine what happens to the function \( f(x) \) as \( x \) gets closer and closer to a particular value, but never quite reaches it, staying slightly less than \( c \). Let's illustrate this with our exercise example: we examined the left-hand limit of the function \( f(x) = -|x| \) as \( x \to 0^- \).

• For \( x < 0 \), the function simplifies to \( f(x) = x \).
• Therefore, as \( x \) approaches 0 from the left, \( f(x) \) approaches 0.

Think of this like coming up to a door from the outside but never quite going in, you're just observing through a window with \( x \) values less than 0.
You note how the behavior or trend of the function appears just before reaching 0.
The insight here is that finding left-hand limits helps us understand one half of the narrative or behavior of a function as it approaches a particular value.

Right-Hand Limit

Right-hand limits are another important concept in calculus that involves approaching a point from values greater than the point. This is denoted as \( \lim_{x \to c^+} f(x) \). It reflects how a function behaves as \( x \) nears a specific value from the right, or from larger values. This can be visualized as approaching a specific feature on a graph from the positive side.
For our exercise with the function \( f(x) = -|x| \), examining the right-hand limit as \( x \to 0^+ \) involves:

• For \( x \ge 0 \), the function becomes \( f(x) = -x \).
• Thus, as \( x \) tends to 0 from the right, \( f(x) \) trends to 0.

This is akin to approaching the same door we mentioned earlier, but this time you're inside the hallway heading toward the door.
Understanding right-hand limits gives insight into another side of the function’s behavior at specific points. By analyzing both left and right-hand limits, mathematicians can grasp the overall situation at a particular point, looking at how both approaches signify the function's continuity or tendency at that juncture.

Absolute Value Function

The absolute value function is a fundamental building block in mathematics, often represented as \( |x| \). It takes any number and returns its distance from zero on a number line. This results in a non-negative output no matter the input sign. In simple terms:

• If \( x \) is positive or zero, \( |x| = x \).
• If \( x \) is negative, \( |x| = -x \).

This creates a V-shaped graph, symmetric along the y-axis when plotted. In the context of the exercise function \( f(x) = -|x| \), it means flipping the absolute value curve upside-down along the x-axis, due to the negative sign outside the absolute value. This affects how the function behaves as \( x \) approaches 0.
The rules of the absolute value ensure that \( -|x| \) effectively considers both the positive and negative tendencies of \( x \). This property is crucial when calculating the left-hand and right-hand limits discussed earlier, anchoring the understanding of changes in displacement whether \( x \) trends toward zero from either direction. Understanding absolute value has broad applications, whether it's assessing real-world measurements or handling equation variables, making it a versatile mathematical concept.