Problem 51 - Calculus and its applications

Question

Graph each function and then find the specified limits. When necessary, state that the limit does not exist. $$ f(x)=|x| ; \quad \text { find } \lim _{x \rightarrow 0} f(x) \text { and } \lim _{x \rightarrow-2} f(x). $$

Step-by-Step Solution

Answer

\( \lim_{x \to 0} f(x) = 0 \) and \( \lim_{x \to -2} f(x) = 2 \).

Step 1: Understanding the Function

The function given is the absolute value function, \( f(x) = |x| \). This function gives the non-negative value of \( x \), which means for \( x \geq 0 \), \( f(x) = x \) and for \( x < 0 \), \( f(x) = -x \).

Step 2: Graphing the Function

The graph of \( f(x) = |x| \) is V-shaped. It has a vertex at the origin (0,0) and consists of two straight lines: \( y = x \) when \( x \geq 0 \) and \( y = -x \) when \( x < 0 \). This is a continuous function without any breaks or holes in the graph.

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

To find \( \lim_{x \to 0} f(x) \), examine the behavior of \( f(x) \) as \( x \) approaches 0 from both sides. From the right, \( x \to 0^+ \), \( f(x) \) approaches 0 since \( f(x) = x \). From the left, \( x \to 0^- \), \( f(x) \) also approaches 0 since \( f(x) = -x \). Both one-sided limits are equal, so \( \lim_{x \to 0} f(x) = 0 \).

Step 4: Finding \( \lim_{x \to -2} f(x) \)

Consider \( f(x) \) as \( x \) approaches \(-2\) from both sides. From the right, \( x \to -2^+ \), \( f(x) = -x \) approaches \( |-2| = 2 \). From the left, \( x \to -2^- \), \( f(x) = -x \) also approaches \( 2 \). Both one-sided limits are the same, thus \( \lim_{x \to -2} f(x) = 2 \).

Key Concepts

Absolute Value Function

The absolute value function, denoted as \( f(x) = |x| \), is a pivotal concept in understanding limits and behavior of functions in calculus. At its core, this function converts all inputs into non-negative outputs. That means whether you input a positive or negative number, the result will always be non-negative.

• If \( x \geq 0 \), then \( f(x) = x \)
• If \( x < 0 \), then \( f(x) = -x \)

This bifurcation ensures that the function behaves like the identity function for non-negative numbers and the negative of the identity for negative numbers. This characteristic creates a unique graphical representation that is crucial when analyzing limits in calculus.

Continuous Function

To describe a continuous function, think of a seamless line you can draw without stopping your pencil. The absolute value function \( f(x) = |x| \) is continuous because you can trace its path across its entire domain without lifting your pen. One of the distinctiveness of continuous functions is their predictability. As you approach any value on the graph, you should be able to confidently determine its exact behavior at that point. This behavior was beautifully illustrated in finding the limits at \( x = 0 \) and \( x = -2 \) for the absolute value function. Since the function does not have any holes, jumps, or asymptotes, its continuous nature is confirmed. Continuous functions like \( f(x) = |x| \) make limit evaluations straightforward, as both sides gather to meet at a single, predictable point.

Graphing Functions

Graphing functions is a powerful tool to visualize mathematical relationships. For the absolute value function \( f(x) = |x| \), the graph is V-shaped. This shape arises because the function pieces together two linear graphs:

• \( y = x \) for \( x \geq 0 \)
• \( y = -x \) for \( x < 0 \)

The point where the two lines meet is called the vertex, located at the origin (0,0). Graphing helps identify such critical points, which are essential when finding limits. When you graph functions, you can visually ascertain how \( f(x) \) behaves as \( x \) approaches specific points, like 0 or -2 in our exercise. This visual insight is instrumental in understanding both the structure and behavior of functions in calculus.