Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \lim _{x \rightarrow-21}(3 x-1)=-64 $$

Step-by-Step Solution

Answer

The limit is \(-64\).

Step 1: Understand the Problem

We need to find the limit of the expression \(3x - 1\) as \(x\) approaches \(-21\). This requires us to substitute the limit value into the expression and simplify.

Step 2: Substitute the Limit Value

Substitute \(x = -21\) into the expression \(3x - 1\). This becomes \(3(-21) - 1\).

Step 3: Simplify the Expression

Calculate \(3 \times (-21) = -63\). Then, subtract 1 from \(-63\) which gives \(-63 - 1 = -64\).

Step 4: Verify the Calculation

Re-examine each step to ensure accuracy. The substitution and arithmetic were done correctly, verifying that the result is \(-64\).

Key Concepts

Limit Evaluation

Limit evaluation is a fundamental concept in calculus that helps us understand the behavior of functions as variables approach specific values. In this exercise, we are tasked to evaluate the limit of the expression \(3x - 1\) as \(x\) approaches \(-21\). Evaluating a limit involves predicting the value that the function is approaching based on the input values. This technique is crucial for dealing with functions that are not easily defined at certain points, such as those involving discontinuities or undefined expressions.

• We start by recognizing the function to be analyzed, here \(3x - 1\).
• The limit requires us to substitute \(x\) with \(-21\) and observe the output.

Grasping the idea of limits enables us to explore more complex equations and their behaviors at specific points.

Substitution Method

The substitution method is a straightforward technique used in calculus to evaluate limits. It involves directly replacing the variable with the given value that it is approaching. In our problem, when we substitute \(x\) with \(-21\) in the expression \(3x - 1\), it turns into \(3(-21) - 1\).

• Substituting the variable directly can often provide an immediate solution, especially when the function remains continuous.
• This method is efficient and clears up how the limits work with simple polynomials.

Remember, substitution is best applied to functions that do not present indeterminate forms when substituted. This concept acts as a precursor to tackling more nuanced limit evaluations.

Algebraic Simplification

Algebraic simplification is the process of making an expression more straightforward to solve. It is particularly important in calculus for limit evaluation because it aids in obtaining a neat and definite result. After substituting \(x = -21\), the expression \(3(-21) - 1\) simplifies by calculating the multiplication and subtraction steps.

• First, solve for \(3 \times (-21)\), which results in \(-63\).
• Next, subtract 1 to get \(-64\).

This simplification confirms that the limit is indeed \(-64\). Breaking down each arithmetic operation step by step ensures that every student can follow along effortlessly.

Calculus Concepts

Calculus is a broad field involving limits, derivatives, integrals, and more. At its core, it allows us to understand and interpret changes and accumulations in functions. This exercise highlights essential calculus concepts such as limits and algebraic manipulation.

• The substitution and simplification showcase foundational calculus techniques.
• Mastering these basics prepares students for more complex concepts like derivatives and integrals.
• Understanding how to evaluate limits forms the basis for everything from continuity to integrating infinite series.

As you continue your calculus journey, keep practicing these basic skills, as they are integral to more advanced topics.