Problem 22 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 1^{-}}(3 x-4)\)

Step-by-Step Solution

Answer

The one-sided limit as x approaches 1 from the negative side of the function \(f(x) = 3x - 4\) exists, and the value of the limit is \(\lim_{x \rightarrow 1^{-}}(3x-4) = -1\).

Step 1: Understand the given function

The function given is a linear function, \(f(x) = 3x - 4\). The graph of this function is a straight line. Linear functions are continuous everywhere, which means that as x approaches any value, the limit from both sides of that value will always exist.

Step 2: Evaluate the limit

Since this is a linear function, we can simply plug in the value of x that we are interested in, which in this case is x=1: \(f(1) = 3(1) - 4 = -1\) \[\lim _{x \rightarrow 1^{-}}(3 x-4) = -1\] As f(x) is continuous everywhere, the limit of the given function as x approaches 1 from the negative side exists, and the value of the limit is -1.

Key Concepts

Linear Functions

When it comes to linear functions, they are essentially the most basic type of algebraic function you can encounter. They are expressed in the form of a straight-line equation, typically represented as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope dictates the steepness of the line and the direction it tilts, while the y-intercept indicates the point where the line crosses the y-axis.

In our particular exercise, the linear function in question is \( f(x) = 3x - 4 \), clearly defining a slope of 3 and a y-intercept of -4. One essential characteristic of linear functions is their predictability and simplicity in calculating values. You can calculate the value of the function for any input \( x \) by using the equation directly, making it straightforward to evaluate limits, as shown in the original solution.

Continuity of Functions

Continuity is a foundational concept in calculus that determines whether a function smoothly transitions from one point to another without any gaps, jumps, or abrupt changes in direction. In more technical terms, a function is considered continuous at a point \( x = a \) if the following three conditions are met:

• The function is defined at \( a \), meaning \( f(a) \) exists.
• The limit of the function as it approaches \( a \) exists.
• The limit of the function as it approaches \( a \) equals \( f(a) \).

Linear functions are inherently continuous over their entire domain because they satisfy these three conditions at every point. This feature allows us to evaluate one-sided limits with the certainty that the function will not exhibit any 'surprises,' such as sudden jumps, at the point of interest.

Limit Evaluation

Limit evaluation is a critical operation in calculus, used to predict a function's behavior as the input approaches a certain value. In the context of our exercise, we're exploring a one-sided limit—specifically, the limit as \( x \) approaches 1 from the left, denoted by \( x \rightarrow 1^{-} \). This notation signifies a scenario where we're only considering the values of \( x \) that are less than 1 (hence coming from the negative side).

The evaluation itself is straightforward for linear functions. Since there's no threat of discontinuity, we can simply substitute the \( x \) value directly into the function to find the limit. If the question asks for a limit that approaches from the negative side, like our problem does, it simply reinforces the value that the function would inherently approach. The step-by-step solution provided is a prime example of the practical application of this concept, with our limit being directly evaluated to \( -1 \) by substitution.