Problem 52 - College Algebra and Calculus: An Applied Approach

Question

find the limit $$ \lim _{x \rightarrow 1} f(x), \text { where } f(x)=\left\\{\begin{array}{ll} x^{2}+2, & x \neq 1 \\ 1, & x=1 \end{array}\right. $$

Step-by-Step Solution

Answer

The limit of the function as \(x\) approaches 1 is 3.

Step 1: Evaluate the Function as x Approaches 1

As \(x\) approaches 1, \(f(x)\) becomes \(1^2 + 2\).

Step 2: Carry out the calculation

Upon substituting, \(f(x)\) equals \(1 + 2 = 3\). This is the value the function approaches as \(x\) tends to 1.

Step 3: Cross-verify with the function definition

The second part of the piecewise function defines \(f(x) = 1\) when \(x=1\). Even though we don't use this part for our limit calculation, it's worth noting to grasp the function's complete behaviour. However, it does not influence the limit.

Key Concepts

Piecewise Functions

Piecewise functions are functions that have different expressions based on distinct intervals of the input variable, often denoted as \(x\). This means that the function behaves differently depending on the domain it is evaluated over. For example, in our given function, \(f(x)=\begin{cases}x^{2}+2, & x eq 1\1, & x=1\end{cases}\), we notice two distinct rules:

• If \(x eq 1\), the function \(f(x) = x^2 + 2\) applies.
• If \(x = 1\), the function is simply \(f(x) = 1\).

These different expressions allow the function to have unique behaviors at specified points. Understanding how piecewise functions work is vital since they enable modeling systems with sudden changes or segments defined by specific conditions.

To properly engage with piecewise functions, consider which part of the function is relevant for the problem at hand, as this dictates what expression to evaluate.

Evaluating Limits

Evaluating limits is a fundamental concept in calculus and involves finding the value that a function approaches as the input draws closer and closer to a given point. In our exercise, we explored the limit of \(f(x)\) as \(x\) approaches 1. Here's how it's done:

• First, identify which expression of the piecewise function is applicable as \(x\) nears 1. Since the limit is considered as \(x \rightarrow 1\) but not exactly equal to 1, the relevant expression to use is \(f(x) = x^2 + 2\).
• Substitute the approaching value of \(x\) into the expression. In our case, we substitute \(x = 1\) into \(x^2 + 2\).
• Calculate the result: \(1^2 + 2 = 3\).

This process shows how, even though \(f(x)\) is defined differently at \(x = 1\), the limit considers the behavior as \(x\) gets close to but does not equal 1.

This concept is key in understanding instantaneous behaviors and changes in calculus, making it a powerful tool for analyzing function limits.

Continuity and Discontinuity

Continuity and discontinuity describe how smoothly a function behaves at a specific point in its domain. A function is continuous at a point if its limit as \(x\) approaches the point is equal to its value at the point.

In the exercise, we have a classic example of discontinuity where the limit of the function as \(x \rightarrow 1\) is 3, while \(f(x) = 1\) when \(x = 1\). When

• The function's limit as \(x \rightarrow 1\) does not equal \(f(1)\).

This results in what's known as a "hole" or "jump" at \(x = 1\), indicating a point of discontinuity. Discontinuities typically represent sudden changes in a function, where there is an unexpected shift in its graph, such as breaking the otherwise smooth line.

Understanding continuity and discontinuity helps in predicting how a function behaves and is especially relevant when modeling real-world situations where abrupt changes occur.