Problem 7 - Calculus: Early Transcendentals

Question

Assume \(\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3,\) and \(\lim _{x \rightarrow 1} h(x)=2 .\) Compute the following limits and state the limit laws used to justify your computations. $$\lim _{x \rightarrow 1}(4 f(x))$$

Step-by-Step Solution

Answer

Question: Find the limit of \(4f(x)\) as \(x\) approaches 1, given that the limit of \(f(x)\) as \(x\) approaches 1 is 8. Answer: The limit of \(4f(x)\) as \(x\) approaches 1 is 32.

Step 1: Write down the given information.

We have the limit of \(f(x)\) as \(x\) approaches 1: \(\lim _{x \rightarrow 1} f(x)=8\).

Step 2: Apply the constant multiple limit law.

According to the constant multiple limit law, we have $$\lim _{x \rightarrow 1}(4 f(x)) = 4 \lim _{x \rightarrow 1} f(x)$$ as the limit of a constant times a function is equal to the constant times the limit of the function.

Step 3: Substitute the limit of \(f(x)\) and find the answer.

Now, we substitute the given limit of \(f(x)\): $$\lim _{x \rightarrow 1}(4 f(x)) = 4 \times 8$$ So, the limit of \(4f(x)\) as \(x\) approaches 1 is $$\lim _{x \rightarrow 1}(4 f(x)) = 32$$.

Key Concepts

Constant Multiple Limit Law

The constant multiple limit law is a fundamental aspect of calculus and is simple yet powerful. It allows us to take a constant factor outside of a limit expression when multiplying it by a function. This means, if you have a constant multiplied by a function, you can first find the limit of the function, and then multiply that result by the constant. This law can be stated as:

• If \( \lim_{x \to a} f(x) = L\), then \( \lim_{x \to a} (c \cdot f(x)) = c \cdot L\)

where \( c \) is a constant.
This principle simplifies limit calculations and is highly useful when dealing with polynomials or products of constants and variables. Applying the law reduces the problem to simply computing the limit of the function involved and multiplying it by the constant. Easy as that!

Limit Computation

Limit computation is all about finding the value that a function approaches as the input gets closer to a certain point. The calculation uses various rules which help in dealing with complex expressions. These rules or laws, such as sum laws, product laws, and the constant multiple limit law, enable us to systematically break down complex limits into something much more manageable.

To compute ends logically:

• Identify the limits involved based on the expressions given.
• Select appropriate limit laws to simplify the expression.
• Substitute the known limits and perform basic arithmetic operations.
• Verify if the limit exists.

Focusing on each step ensures clarity and avoids common pitfalls in limit calculations. If you maintain consistency with the limit laws, calculating limits becomes straightforward and intuitive.

Functions Limits

Understanding the behavior of functions as they approach certain points is an essential skill in calculus. The limit of a function describes what value a function approaches as its input approaches a specific point. Calculating limits helps in understanding continuity, and in the analysis of derivatives and integrals, as it describes how a function behaves around critical points.
Common practices in calculating function limits involve:

• Using algebraic manipulation to simplify the function.
• Approximating the behavior of the function around a given point.
• Applying limit laws to determine the precise value the function approaches.

Function limits form the foundation of calculus, offering insights into dynamic systems, allowing predictions about future behavior, and providing a basis for more advanced topics such as differential calculus.