Problem 19 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In Problems 19 and 20, give an example of a function having the set of characteristics specified. (a) \(\lim _{x \rightarrow 5} f(x)=7 ; f(5)=7\) (b) \(\lim _{x \rightarrow 5} g(x)=7 ; g(5)=8\)

Step-by-Step Solution

Answer

a) The function \(f(x) = 7\) has \(\lim _{x \rightarrow 5} f(x)=7\) and \(f(5)=7\). b) The piecewise defined function \(g(x) = 8\) if \(x = 5\) and \(g(x) = 7\) otherwise has \(\lim _{x \rightarrow 5} g(x)=7\) and \(g(5)=8\).

Step 1: Find a function for (a)

We need to find a function where the limit as x approaches 5 is 7 and the function value at x = 5 is also 7. The function \(f(x) = 7\) fulfills these conditions. The limit as x approaches any value is always 7, as is the value of the function at any point.

Step 2: Test the function for (a)

Substituting x=5 into \(f(x) = 7\) gives f(5)=7. The limit of the function as x approaches 5 can be calculated as \(\lim _{x \rightarrow 5} 7 = 7\). Both the limit and the function value at x=5 are 7, so \(f(x) = 7\) is a correct answer.

Step 3: Find a function for (b)

We need to find a function where the limit as x approaches 5 is 7, but the function value at x = 5 is 8. One example of a function that fulfills these conditions is the piecewise defined function \(g(x) = 8\) if \(x = 5\) and \(g(x) = 7\) otherwise.

Step 4: Test the function for (b)

The value of the function at x=5 is defined to be 8, so \(g(5) = 8\). The limit of the function as x approaches 5 is 7, because that is the function value at all other points. Therefore, \(g(X) = 8\) if \(x = 5\) and \(g(x) = 7\) otherwise is a correct answer.

Key Concepts

Calculus

Calculus, a branch of mathematics rich in applications and conceptual frameworks, is fundamentally about change and motion. It's split into two main areas known as differential calculus and integral calculus, which are concerned with rates of change and accumulation of quantities, respectively. In problems involving calculus, you often examine functions and their behavior. Such an analysis includes finding limits, which highlight the behavior of functions as inputs approach certain values, and is pivotal in understanding how these changes occur. For instance, understanding the limit of a function as the input nears a particular point helps in solving problems related to instantaneous rates of change—essential in physics and engineering.

Piecewise Functions

Piecewise functions are like mathematical chameleons, changing their formula based on the input value. They're defined by multiple sub-functions, each applying to a certain interval or point in their domain. This makes them incredibly useful in modeling scenarios where a situation changes afta certainthresd or conditions. The versatility of piecewise functions allows them to depict a range of phenomena, from tax brackets to physical processes that shift behavior at certain thresholds. When dealing with limits and continuity, piecewise functions can demonstrate peculiar traits, as their behavior at the junctions of sub-functions can vary significantly from one side to the other.

Continuity

Continuity in calculus is akin to a smooth journey without any sudden jumps or breaks. A function is considered continuous at a point if there is no interruption in its graph at that point. This means several things: the function is defined at the point, the limit exists as you approach the point from all directions, and the limit equals the function's value at that point. Through this, continuity ensures that small changes in the input result in small changes in the output, a property crucial in predicting and understanding real-world behaviors. When functions are continuous across their entire domain, they're easier to analyze and integrate, making continuity a desirable trait in functions used in modeling and problem-solving.

Limit Definition

The limit of a function is a fundamental concept in calculus that describes the behavior of that function as the input approaches a particular value. The formal definition of a limit is rather technical, but in essence, it articulates what value the function outputs get infinitely close to as the input gets infinitely close to some point, without necessarily reaching that point. Limits help bypass mathematical indeterminacy, allow the examination of function values at points of discontinuity, and are crucial for defining derivatives and integrals. Understanding the limit concept is pivotal for tackling a myriad of problems in calculus, as well as in the fields that rely on its principles, such as physics, engineering, and economics.