Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Find the limit. $$ \lim _{x \rightarrow-3}(2 x+5) $$

Step-by-Step Solution

Answer

The limit of the function \(2x+5\) as \(x\) approaches -3 is -1.

Step 1: Identify the Function

The function we’re working with is \(2x+5\). We’re finding the limit as \(x\) approaches -3. This is a simple linear function, and there are no divide-by-zero issues or other discontinuities, so we can simply substitute -3 for \(x\).

Step 2: Substitute

We substitute -3 for \(x\) in the given function. So, \(2(-3) + 5 = -6 + 5 = -1\).

Key Concepts

Understanding Calculus and Limits

Calculus is a fundamental branch of mathematics that explores change and motion. In this field, one of the essential concepts is that of limits. Limits help us understand the behavior of functions as variables approach a certain value. For example, when solving the question Find the limit: \(lim _{x \rightarrow-3}(2 x+5)\), we're trying to determine what value the linear function \(2x + 5\) approaches as \(x\) gets closer and closer to -3.

Understanding limits is crucial because they form the foundation of more complex topics in calculus, such as derivatives and integrals. These concepts are all connected by the notion of approaching a particular point, either on a graph or numerically, and analyzing the outcome. The steps provided in our exercise demonstrate how to directly evaluate simple limits by substitution. Also, a limit can tell us about the value a function is nearing, without the function actually reaching that point, which is particularly useful in cases of discontinuity or undefined expressions.

The Simplicity of Linear Functions

A linear function, such as \(2x + 5\), is one of the simplest forms of a function. It represents a straight line on a graph and is defined by an equation in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In our exercise, the slope \(m\) is 2, indicating the steepness of the line, and the y-intercept \(b\) is 5, showing where the line crosses the y-axis.

One of the main characteristics of a linear function is its constant rate of change. This means that for any given change in \(x\), the change in \(y\) is always the same, making the graph of these functions a straight line. This property also simplifies the process of finding limits because we expect the function to approach a single predictable value as \(x\) approaches any particular number, just as we saw with the direct substitution done in Step 2 of the solution.

Continuity of Functions

Continuity is a property of functions that tells us whether a function's graph can be drawn without lifting the pencil off the paper. For a function to be continuous at a point, it must meet three criteria: the function must be defined at the point, the limit as \(x\) approaches the point from both directions must exist, and the limit must equal the function's value at that point.

In our problem, the function \(2x+5\) is continuous because it meets all these criteria. When we find the limit as \(x\) approaches -3, both the left-hand limit and the right-hand limit are equal to -1, and the function is defined at \(x = -3\). Therefore, there is no interruption or 'jump' in the graph at this point. This seamless behavior confirms continuity and is expected for all linear functions over their entire domain. By understanding the concept of continuity, students can better analyze and graph various types of functions as they progress in the study of calculus.