Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \lim _{x \rightarrow 1}\left(2 x^{2}+1\right)=3 $$

Step-by-Step Solution

Answer

The limit is 3.

Step 1: Understand the Limit Expression

The expression \( \lim _{x \rightarrow 1}\left(2 x^{2}+1\right) \) is asking for the value that \( 2x^2 + 1 \) approaches as \( x \) gets closer and closer to 1.

Step 2: Substitute the Value into the Expression

Substitute \( x = 1 \) into the expression \( 2x^2 + 1 \). This means calculating \( 2(1)^2 + 1 \).

Step 3: Calculate the Expression

Calculate \( 2(1)^2 + 1 \) which simplifies to \( 2 \times 1 + 1 = 2 + 1 = 3 \).

Step 4: Confirm the Limit Value

Since substituting \( x = 1 \) into the expression results in \( 3 \), the value \( 2x^2 + 1 \) approaches is indeed \( 3 \) as \( x \) approaches \( 1 \). Therefore, \( \lim _{x \rightarrow 1}\left(2 x^{2}+1\right) = 3 \).

Key Concepts

Limit Evaluation

In calculus, a limit evaluates the behavior of a function as the input approaches a certain value. For example, in the expression \( \lim _{x \rightarrow 1}\left(2 x^{2}+1\right) \), we want to determine what value the expression \( 2x^2 + 1 \) reaches as \( x \) gets closer and closer to 1. This process helps us understand more about the function's continuity and potential behavior around that specific point. Limits are fundamental to understanding derivatives and integrals, making them a cornerstone of calculus.
Evaluating a limit can sometimes indicate whether a function approaches a finite value, infinity, or remains undefined at a certain point.
Overall, proper evaluation of limits provides insights into the function's overall behavior.

Substitution Method

The substitution method is a straightforward technique to evaluate limits. It involves directly substituting the value that \( x \) is approaching, into the function. In the given exercise, the substitution method is applied by replacing \( x \) with 1 in the expression \( 2x^2 + 1 \).
Steps for using the substitution method include:

• Identify the value that \( x \) approaches, which is 1 in this scenario.
• Substitute this value into the function, simplifying if necessary.
• Calculate to find the resulting value of the expression.

This method works well when the function is continuous at the point being evaluated. If the evaluated value makes the function undefined or results in an indeterminate form like \( \frac{0}{0} \), other more advanced techniques may be used, such as factoring or L'Hopital's Rule.

Calculus Fundamentals

Calculus is the mathematical study of continuous change, central to which are the concepts of limits, derivatives, and integrals. The idea of limits provides the foundation for defining derivatives and integrals.
Understanding calculus begins with mastering the fundamentals:

• Limits: The basic tool for finding predictions about function behaviors, crucial for defining continuous functions and understanding change.
• Derivatives: These describe how a function changes as its input changes, providing rates of change and slope of the function graph, being an extension of the concept of limits.
• Integrals: Integrals accumulate values across a range, providing another essential way of understanding functions and areas under curves.

Grasping these core elements allows students to understand more complex concepts and solve practical problems in physics, engineering, and other fields. It's recommended to practice plenty of problems to build a strong foundation in these areas.