Problem 42 - Calculus for Biology and Medicine

Question

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 2}\left(8 x^{3}-2 x+4\right) $$

Step-by-Step Solution

Answer

The limit is 64.

Step 1: Identify the Limit Expression

Given the expression, identify the components involved. We are tasked to find \( \lim _{x \rightarrow 2}(8x^3 - 2x + 4) \).

Step 2: Apply Direct Substitution

Use the direct substitution method by plugging the value of \( x = 2 \) directly into the polynomial expression since polynomials are continuous functions. Therefore, compute \( 8(2)^3 - 2(2) + 4 \).

Step 3: Calculate Each Term Separately

Calculate each term separately: \( 8(2)^3 = 64 \), \( 2(2) = 4 \), and the constant term is \( 4 \).

Step 4: Combine the Results

Combine the results from the previous step: \( 64 - 4 + 4 \).

Step 5: Simplify the Expression

Simplify the expression: First, add and subtract as needed to simplify \( 64 - 4 + 4 = 64 \).

Step 6: Conclude the Limit Evaluation

With all calculations completed, conclude that the limit as \( x \) approaches 2 is 64.

Key Concepts

Limit Laws

When you're working with limits, it's important to understand the foundational concepts that make solving these problems much easier. Limit laws are like guidelines or shortcuts that allow us to calculate limits efficiently. They apply to a variety of operations, such as adding, subtracting, and multiplying functions.
For instance, if you know the limits of two separate functions, you can add them just like regular numbers using the limit law for addition. Here are some essential limit laws you should know:

• Sum Law: The limit of the sum of two functions is the sum of their limits. Mathematically, if you have \( ext{lim}_{x \to a} [f(x) + g(x)] = ext{lim}_{x \to a}f(x) + ext{lim}_{x \to a}g(x) \).
• Difference Law: Similar to the Sum Law, the limit of the difference is the difference of their limits: \( ext{lim}_{x \to a} [f(x) - g(x)] = ext{lim}_{x \to a}f(x) - ext{lim}_{x \to a}g(x) \).
• Product Law: The limit of a product of two functions is the product of their limits. If \( ext{lim}_{x \to a} [f(x) \, g(x)] = ext{lim}_{x \to a}f(x) \, \cdot \, \text{lim}_{x \to a}g(x) \).
• Constant Multiple Law: The limit of a constant times a function is the constant times the limit of the function. It looks like \( ext{lim}_{x \to a} [c \, f(x)] = c \, ext{lim}_{x \to a}f(x) \).

These laws simplify evaluating limits and particularly come in handy when dealing with more complex expressions, where you could break them down into simpler parts.

Direct Substitution Method

The direct substitution method is one of the simplest and most frequently used techniques to find limits, especially when working with polynomials. Since polynomials are continuous functions, they allow us to substitute the limit point directly into the polynomial expression. This makes it easy to evaluate the limit as the function value approaches the point.
Consider a polynomial like \(8x^3 - 2x + 4\). Here’s how you use the direct substitution method:

• Substitute the limit point directly into the polynomial: Replace \(x\) with the value it is approaching, which in our example is 2.
• Perform the arithmetic operations as usual, simplifying the expression step by step.
• After simplifying, the result is the limit as \(x\) approaches the chosen point.

By directly inputting the value and simplifying, we essentially "follow the path" that the function takes as \(x\) approaches our selected point, and we find the output without much algebraic manipulation. It is a straightforward and effective approach that works nicely due to the continuous nature of polynomial functions.

Polynomial Continuity

Polynomials are known for their smooth and continuous nature across their entire domain. This quality, known as continuity, means there are no breaks, jumps, or holes in the graph of a polynomial. It simplifies the process of finding limits significantly.
When we say a polynomial is continuous for all real numbers, we mean you can draw the entire graph without lifting your pencil. Because of this, evaluating limits with polynomials often involves direct substitution. Here are some reasons why polynomials are friendly for limit evaluations:

• Continuity: Polynomials have no points of discontinuity. This means limits can be evaluated right at the point of interest, making them perfect candidates for the direct substitution method.
• Simple Arithmetic: The operations involved in polynomials (addition, subtraction, multiplication) comply nicely with direct substitution and limit laws.
• Predictable Behavior: As \(x\) approaches a certain point, the behavior of polynomials is predictable and follows the same smooth curve pattern.

Overall, the inherent continuity of polynomials makes evaluating limits an easy task. By understanding this concept, you can readily solve limit problems by plugging in the specific value directly into the polynomial. This characteristic sets polynomials apart in calculus, making them more approachable and less complex to handle in limit scenarios.