Problem 70 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer. The limit of the product of two functions is equal to the product of the limits of the two functions.

Step-by-Step Solution

Answer

The statement is true, providing that the limits of the functions exist.

Step 1: Introduce the Product Limit Law

In calculus, the Product Limit Law states that the limit of the product of two functions is the product of their limits, given the limits of the two functions exist.Here is the formal representation of the Product Limit Law: If \( f \) and \( g \) are functions such that \(\lim_{{x \to a}}f(x)\) = \(L\) and \(\lim_{{x \to a}}g(x)\) = \(M\), then \(\lim_{{x \to a}}[f(x)g(x)]\) = \(LM\).

Step 2: Apply the theorem to the statement

The given statement says: 'The limit of the product of two functions is equal to the product of the limits of the two functions'. This is indeed the same as the Product Limit Law, which means the statement is true, provided the limits of the functions exist.

Key Concepts

Limits of Functions

Understanding the concept of limits of functions is foundational to studying calculus. A limit attempts to describe the behavior of a function as its input approaches a certain value, but it is not focused on the function's value at that point itself. Imagine, for instance, walking towards a wall; as you get closer, which represents the input value approaching a particular point, you can describe your proximity to the wall, even if you never actually touch it.

Mathematically, the notation \( \lim_{{x \to a}}f(x) \) signifies the limit of function \( f \) as \( x \) approaches the value \( a \) from either direction on the number line. The final value, which \( f(x) \) approaches but does not necessarily reach, is what we call the limit. It's essential to note that the limit may exist even if the function is undefined or takes on a different value at \( a \) itself. Calculating limits is crucial for analyzing how functions behave near specific points, and thus integral to understanding subsequent concepts in calculus.

Calculus

The field of calculus is a branch of mathematics that studies continuous change. It's divided primarily into two subfields: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which measures the rate at which a quantity changes. Conversely, integral calculus is concerned with the accumulation of quantities, such as areas under curves, expressed through integrals.

Deeply intertwined with concepts of limits, calculus allows us to understand aspects of motion, growth, and a variety of other phenomena in the physical world. It underpins much of modern science and engineering. The Product Limit Law, which asserts that the limit of a product is the product of the limits, if those limits exist, is one piece of the larger puzzle in calculus. By breaking down complex problems into simpler parts, calculus provides us with a powerful toolkit for solving both theoretical problems and practical, real-world challenges.

Functional Limits

The concept of functional limits is specifically about understanding the behavior of functions as inputs near a specific value. In the context of the Product Limit Law, we look at the limits of two separate functions, say \( f \) and \( g \), as they independently approach their limits as \( x \) approaches a value \( a \).

It's crucial to ascertain that the individual limits exist when analyzing functional limits. If that's the case, as per the Product Limit Law, the limit of the product of these two functions as \( x \) approaches \( a \) will be the product of their respective limits. This is largely useful when dealing with composite functions or when simplifying complex expressions by breaking them down into more manageable components. Functional limits are pivotal for predicting and understanding the behavior of complex systems in various scientific fields.