Problem 56 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f\) is continuous at \(x=a\) and \(g\) is differentiable at \(x=a\), then \(\lim _{x \rightarrow a} f(x) g(x)=f(a) g(a)\).

Step-by-Step Solution

Answer

The statement is true. Since \(f\) is continuous at \(x=a\) and \(g\) is differentiable at \(x=a\) (implying continuity), their product is also continuous at \(x=a\). Therefore, \(\lim_{x \rightarrow a} f(x)g(x) = \lim_{x \rightarrow a} f(x) \cdot \lim_{x \rightarrow a} g(x) = f(a) \cdot g(a)\).

Step 1: Analyze continuity at \(x=a\)#

As given, \(f\) is continuous at \(x=a\). By definition, this means \(\lim_{x \rightarrow a} f(x) = f(a)\). This property will be used later as we progress through the solution.

Step 2: Analyze differentiability at \(x=a\)#

We are given that \(g\) is differentiable at \(x=a\). This means that the derivative \(g'(x)\) exists at \(x=a\). Additionally, differentiability implies continuity, so we can also conclude that \(g\) is continuous at \(x=a\). Hence, \(\lim_{x \rightarrow a} g(x) = g(a)\).

Step 3: Calculate the limit of the product#

Now, we need to find the limit of the product of the two functions as \(x\) approaches \(a\): \(\lim_{x \rightarrow a} f(x)g(x)\). We note that since both \(f\) and \(g\) are continuous at \(x=a\), their product should also be continuous at \(x=a\). Using the limit properties for continuous functions, and the limits we found in the previous steps, we have: \(\lim_{x \rightarrow a} f(x)g(x) = \lim_{x \rightarrow a} f(x) \cdot \lim_{x \rightarrow a} g(x) = f(a) \cdot g(a)\). Thus, the statement \(\lim_{x \rightarrow a} f(x) g(x)=f(a) g(a)\) is true, as shown by using the properties of continuous functions and differentiable functions.

Key Concepts

Understanding Continuity

In calculus, continuity is a crucial concept that helps us understand how functions behave as they approach a certain point. A function \( f \) is said to be continuous at a point \( x = a \) if the following condition holds:

• \( \lim_{x \rightarrow a} f(x) = f(a) \)

This equation indicates that as \( x \) approaches \( a \), the function \( f(x) \) approaches the value \( f(a) \). There is no sudden jump or gap in the graph of the function at \( x = a \).
Continuity is important because it ensures that small changes in \( x \) result in small changes in \( f(x) \). This predictable behavior is foundational in calculus, allowing us to apply limits and perform further actions like integration and differentiation with confidence.
When analyzing the product of two functions, if both are continuous at a point, then their product is also continuous at that point. This property makes it easy to analyze combinations of functions.

The Concept of Differentiability

Differentiability refers to the ability of a function to have a derivative at a certain point. For a function \( g \) to be differentiable at \( x = a \), the derivative \( g'(x) \) must exist at that point.
If a function is differentiable at a point, it is also continuous at that point. This is because differentiability is a stronger condition than continuity. It means the function not only lacks gaps or jumps but also changes smoothly without any sharp turns or breaks.
Let's summarize the key aspects of differentiability:

• A function is differentiable at \( x = a \) if \( g'(a) \) exists.
• Differentiability implies continuity.
• If a function is not differentiable, it may still be continuous but will lack a smooth tangent at that point.

Understanding whether a function is differentiable helps in constructing more complex analyses, such as examining the behavior of entire function families. For the statement in the exercise, ensuring differentiability helps reinforce the implied continuity, enabling the use of product limits.

Limits of Functions

The concept of limits is central to calculus. It allows us to understand the behavior of functions as they approach a particular value. A limit expresses what a function approaches as the input, \( x \) approaches a certain point.
Consider \( \lim_{x \rightarrow a} f(x) = L \), meaning as \( x \) gets closer to \( a \), \( f(x) \) tends towards \( L \). Limits are foundational to defining continuity and differentiability.

• When using limits to show continuity, we need \( \lim_{x \rightarrow a} f(x) = f(a) \).
• For differentiability, the limit involved is defined in the derivative \( f'(a) = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h} \).

In the statement from the exercise, finding the limit of the product \( \lim_{x \rightarrow a} f(x)g(x) \) relies on the individual limits of \( f(x) \) and \( g(x) \) being equal to their values at \( a \), thanks to continuity. It demonstrates how individual limits allow us to derive more complex limits, leading to a rich understanding of function behavior.