Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises 21-38, find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 1^{+}}(2 x+4)\)

Step-by-Step Solution

Answer

The right-sided limit of the function \(f(x) = 2x + 4\) as \(x\) approaches 1 is 6.

Step 1: Identify the Function and the Point

Here, we have the function \(f(x) = 2x + 4\) and we're tasked with finding the limit as \(x\) approaches 1 from the right side, denoted as \(x \to 1^+\).

Step 2: Check for Discontinuity

The function \(f(x) = 2x + 4\) is a linear function, thus it's continuous everywhere. Therefore, there are no discontinuities at \(x=1\).

Step 3: Find the Right-Sided Limit

To find the limit as \(x\) approaches 1 from the right side, we will simply substitute 1 into the function: \[ \lim _{x \rightarrow 1^{+}}(2x+4) = 2(1) + 4 = 2 + 4 \]

Step 4: Evaluate the Limit

Now, simplify the expression to get the final result: \[ \lim _{x \rightarrow 1^{+}}(2x+4) = 2 + 4 = 6 \] The right-sided limit of the function \(f(x) = 2x + 4\) as \(x\) approaches 1 is 6.

Key Concepts

Limit of a Function

Understanding the limit of a function is crucial when studying calculus as it deals with the behavior of a function as it approaches a certain point. In the scenario provided, we are focused on a one-sided limit, specifically a right-sided limit, denoted as \( \lim _{x \rightarrow a^{+}}f(x) \), which means we are looking at the value the function approaches as the variable \( x \) gets infinitely close to the number \( a \) from the right (increasing \( x values \)).

In the example of \( \lim _{x \rightarrow 1^{+}}(2x+4) \), we substitute \( x \) with 1 because the linear function \( f(x) = 2x + 4 \) is continuous and doesn't have any discontinuities near \( x=1 \). When we evaluate the expression, we get 6. This means as we get closer and closer to 1 from the right, the value of the function gets closer to 6.

Continuous Function

A continuous function is a function that has no abrupt changes or gaps in its graph. In simpler terms, you can draw the function without lifting your pencil off the paper. Continuity is essential because it ensures that the function behaves predictably around the point in question and the limit exists. For a function to be continuous at a specific point, three conditions must be met: the function must be defined at that point, a limit must exist at that point, and the limit must be equal to the function's value at that point.

In our exercise \( f(x) = 2x + 4 \), the graph is a straight line, which means it is a model example of a continuous function. There are no holes, jumps, or vertical asymptotes, so the limit as \( x \) approaches any value is simply the function's value at that point. Therefore, when you are asked to find the limit of a continuous function like a linear function, you can often just plug in the value of \( x \) to get the answer.

Linear Function

A linear function is one of the simplest forms of functions where the graph is a straight line, and the general form of the equation is \( f(x) = mx + b \), with \( m \) being the slope and \( b \) being the y-intercept. Linear functions are always continuous because they do not have breaks or bends in their graphs. In the context of the given exercise, \( f(x) = 2x + 4 \) is a linear function with a slope of 2 and a y-intercept of 4.

The beauty of linear functions lies in their predictability and simplicity. When we calculate the limit of a linear function, as \( x \) approaches any real number, we can consistently find that this limit is equal to the actual value of the function at that point. Therefore, understanding linear functions is not only essential for conceptual clarity, but it also makes solving limit problems more intuitive.