Problem 62 - Calculus and its applications

Question

Graph each function and then find the specified limits. When necessary, state that the limit does not exist. $$ \begin{array}{l} G(x)=\left\\{\begin{array}{ll} -x+3, & \text { for } x<2 \\ x+1, & \text { for } x \geq 2 \end{array}\right. \\ \text { Find } \lim _{x \rightarrow 2^{-}} G(x), \lim _{x \rightarrow 2^{+}} G(x), \text { and } \lim _{x \rightarrow 2} G(x) . \end{array} $$

Step-by-Step Solution

Answer

\( \lim_{x \to 2^{-}} G(x) = 1 \), \( \lim_{x \to 2^{+}} G(x) = 3 \), and \( \lim_{x \to 2} G(x) \) does not exist.

Step 1: Understand the Function

The function \( G(x) \) is a piecewise function, meaning it has different expressions based on the value of \( x \). For \( x < 2 \), \( G(x) = -x + 3 \). For \( x \geq 2 \), \( G(x) = x + 1 \). We need to graph \( G(x) \) to understand its behavior around \( x = 2 \).

Step 2: Graph the Function for \( x < 2 \)

For the segment where \( x < 2 \), the function is \( G(x) = -x + 3 \). This is a straight line with a slope of -1 and a y-intercept at 3. Plugging specific values such as \( x = 0 \), \( G(0) = 3 \) and \( x = 1 \), \( G(1) = 2 \), helps in sketching the graph.

Step 3: Graph the Function for \( x \geq 2 \)

For \( x \geq 2 \), \( G(x) = x + 1 \). This is also a straight line with a slope of 1 and a y-intercept at 1. Starting at \( x = 2 \), \( G(2) = 3 \), we use this to continue the line for \( x > 2 \), checking points like \( x = 3 \), \( G(3) = 4 \).

Step 4: Determine \( \lim_{x \to 2^{-}} G(x) \)

The left-hand limit, \( \lim_{x \to 2^{-}} G(x) \), approaches the value on the graph as \( x \) gets close to 2 from the left. With the expression \(-x + 3\), calculate \( \lim_{x \to 2^{-}} (-x + 3) = -2 + 3 = 1 \).

Step 5: Determine \( \lim_{x \to 2^{+}} G(x) \)

The right-hand limit, \( \lim_{x \to 2^{+}} G(x) \), considers the value as \( x \) approaches 2 from the right with \( x + 1 \). This gives \( \lim_{x \to 2^{+}} (x + 1) = 2 + 1 = 3 \).

Step 6: Determine \( \lim_{x \to 2} G(x) \)

Both the left-hand limit and right-hand limit must match for the overall limit \( \lim_{x \to 2} G(x) \) to exist. Since \( \lim_{x \to 2^{-}} G(x) = 1 \) and \( \lim_{x \to 2^{+}} G(x) = 3 \), the limits do not match, meaning \( \lim_{x \to 2} G(x) \) does not exist.

Key Concepts

Understanding Piecewise Functions

A piecewise function is composed of multiple sub-functions, each of which is defined over a specific interval. It allows us to describe functions with different behaviors over different ranges. For instance, the function \( G(x) \) in your exercise is a classic example of a piecewise function, defined as:

• \( G(x) = -x + 3 \) for \( x < 2 \)
• \( G(x) = x + 1 \) for \( x \geq 2 \)

This means that depending on the value of \( x \), you use different equations. For \( x < 2 \), the function behaves according to \( -x + 3 \); once \( x \) reaches 2 or goes beyond, the function switches to the \( x + 1 \) expression. Recognizing the conditions attached to each piece is crucial when dealing with such functions. They are useful in various applications, particularly when dealing with scenarios that change abruptly at certain points like tax calculations or speed zones in driving.

The Concept of Continuity

Continuity is an important concept that helps us understand whether a function behaves without any interruptions or jumps at a certain point. For a function to be continuous at a point, it must satisfy three conditions:

• The function must be defined at that point.
• The limit of the function as it approaches the point from both left and right must exist.
• The left-hand limit, right-hand limit, and the function's value at that point must all be equal.

In the context of your exercise, examining the limits as \( x \to 2 \) is key. For \( G(x) \), the left-hand limit \( \lim_{x \to 2^{-}} G(x) = 1 \) and the right-hand limit \( \lim_{x \to 2^{+}} G(x) = 3 \). Since these do not equal each other, \( G(x) \) is not continuous at \( x = 2 \). This "jump" indicates a discontinuity, which we often use to identify different types of changes in real-world processes.

Graphing Functions Accurately

Graphing functions, especially piecewise ones, is a valuable skill in calculus. It allows us to picture how the function behaves at every interval. When graphing piecewise functions like \( G(x) \), it's important to:

• Highlight the intervals and equations that define the function for each segment on your graph.
• Mark closed or open dots at points where functions change or connect, indicating whether the value at that point is included or not.

For example, for \( G(x) \) at \( x = 2 \), there would be a filled dot at the point \((2, 3)\) for the function \( x + 1 \) and an open circle at \((2, 1)\) for the function \( -x + 3 \) to denote that this value is not included. This graphical representation helps in visualizing what happens as \( x \) approaches 2 from both sides.
Such visualization not only aids in understanding but also ensures that you accurately describe any jumps or discontinuities present in the function. Understanding how to transition between different segments on the graph enhances your ability to analyze and interpret functions effectively.