Problem 68 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(67 - 70\) , use the following function. \(f ( x ) = \left\\{ \begin{array} { l l } { 2 - x , } & { x \leq 1 } \\ { \frac { x } { 2 } + 1 , } & { x > 1 } \end{array} \right.\) Multiple Choice What is the value of \(\lim _ { x \rightarrow 1 } + f ( x ) ?\) (A) 5\(/ 2\) \(( \mathrm { B } ) 3 / 2\) \(( \mathbf { C } ) 1\) \(( \mathbf { D } ) 0\) (E) does not exist

Step-by-Step Solution

Answer

(B) 3/2

Step 1: Identify the Relevant Function Definition

Since the question asks for the limit as x approaches 1 from the right (\(x \rightarrow 1 ^+ \)), examine the definition of f(x) for \(x > 1\), which is \(x/2 + 1\).

Step 2: Substitute Limit Value

Next, substitute the value of x=1 into the function definition, so it becomes \(1/2 + 1 = 3/2\).

Step 3: Conclude the Result

The value obtained is the limit of the function as \(x \rightarrow 1 ^+ \). Thus, \( \lim _ { x \rightarrow 1 } + f ( x ) = 3/2 \)

Key Concepts

Limits

In calculus, limits help us understand the behavior of a function as its input approaches a certain value. It's like trying to find out what happens to the function's output as we get really close to a specific input value. Limits are a foundational concept because they are used to define derivatives and integrals, which are crucial to calculus. Here are a few key points to consider about limits:

• They help describe how functions behave at boundary points.
• Limits look at the value a function approaches, not necessarily the value the function actually takes at that point.
• They are used to determine the continuity of a function, among many other things.

When you're solving a problem involving limits, sometimes it involves simple substitution, but often it's about analyzing the function's behavior near the point of interest.
It's important to note that limits can be one-sided, meaning they can approach the point from the left or the right.

Piecewise Functions

Piecewise functions are those that have different expressions for different intervals of their domain. In simpler terms, it is one function that is broken into multiple parts, and each part has its expression or rule depending on the value of the input. Here’s how piecewise functions are structured:

• They consist of two or more "pieces," each with its own function rule.
• Each piece applies to a specific interval of the input values.
• The function can be continuous or discontinuous depending on how the pieces connect at the endpoints of their intervals.

For example, the function given in the exercise is a piecewise function where the rule changes at the point where the input is 1. Understanding how these pieces fit together is essential for determining limits, derivatives, and integrals involving piecewise functions.

Right-Hand Limit

The right-hand limit refers to understanding what happens to a function's output as the input approaches a certain point from the right, or greater values. It's a part of studying one-sided limits, where the input approaches the specific value from one direction only. Here's how the right-hand limit is analyzed:

• Identify which part of the piecewise function applies as the input approaches the point from the right.
• Plug the point value into this piece's expression if applicable, or analyze the limit behavior.
• The right-hand limit is particularly useful in determining the continuity and differentiability at points where the function definition changes.

In the example given, the right-hand limit as the input approaches 1 involves using the function piece defined for when the input is greater than 1. By understanding right-hand limits, you learn how to evaluate changes and breakpoints in piecewise functions accurately.