Problem 2 - A Graphical Approach to Precalculus with Limits

Question

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4}[g(x) \cdot f(x)]$$

Step-by-Step Solution

Answer

The limit is 128.

Step 1: Identify the Given Information

We know that the limit of \(f(x)\) as \(x\) approaches 4 is 16, i.e., \(\lim_{x \rightarrow 4} f(x) = 16\). Similarly, the limit of \(g(x)\) as \(x\) approaches 4 is 8, i.e., \(\lim_{x \rightarrow 4} g(x) = 8\). We need to find the limit of the product \(g(x) \cdot f(x)\) as \(x\) approaches 4.

Step 2: Apply the Product Rule for Limits

The limit of the product of two functions as \(x\) approaches a certain value can be found by multiplying the limits of the individual functions. So, we can use the product rule of limits which states: \[\lim_{x \rightarrow a} [f(x) \cdot g(x)] = \left(\lim_{x \rightarrow a} f(x)\right) \cdot \left(\lim_{x \rightarrow a} g(x)\right)\]

Step 3: Substitute the Given Limits into the Product

Substitute the known limits into the product equation:\[\lim_{x \rightarrow 4} [g(x) \cdot f(x)] = (16) \cdot (8)\]

Step 4: Perform the Multiplication

Now, multiply the two limits: \[16 \cdot 8 = 128\]

Step 5: Write the Final Answer

Based on the multiplication operation, the limit is:\[\lim_{x \rightarrow 4} [g(x) \cdot f(x)] = 128\]

Key Concepts

Product Rule for Limits

When dealing with limits, especially when multiplying two functions together, one of the key tools is the Product Rule for Limits. This rule simplifies finding the limit of a product by letting you find each individual limit first.

The rule states that if you have two functions, say \( f(x) \) and \( g(x) \), and you know their limits as \( x \) approaches a certain value, then the limit of their product is simply the product of the two individual limits.

Mathematically, the rule can be expressed as:
\[\lim_{x \rightarrow a} [f(x) \cdot g(x)] = \left(\lim_{x \rightarrow a} f(x)\right) \cdot \left(\lim_{x \rightarrow a} g(x)\right)\]

For example, if \( \lim_{x \rightarrow a} f(x) = L \) and \( \lim_{x \rightarrow a} g(x) = M \), then \( \lim_{x \rightarrow a} [f(x) \cdot g(x)] = L \cdot M \). This greatly simplifies many limit problems by breaking them down into more manageable pieces.

Limit Evaluation

Limit evaluation is a technique used to find what value a function approaches as the variable gets closer to a specific point. It’s fundamental in calculus and helps us understand the behavior of functions near certain values, even if the function doesn't explicitly reach that value.

When approaching a limit problem, follow these steps:

• Identify the target point, i.e., the value that \( x \) is approaching.
• Find the limit of the function by substituting the point into the function if it doesn’t cause undefined operations like division by zero.
• If substitution isn’t possible, use algebraic manipulations or the limit laws to find the limit.

In the case of \( \lim_{x \rightarrow 4} f(x) \) and \( \lim_{x \rightarrow 4} g(x) \), we can directly substitute the limits given as 16 and 8, respectively, without further manipulation, as these values don’t lead to undefined expressions.

Function Multiplication

Function multiplication involves multiplying the outputs of two functions. It's an important concept in calculus because it shows how changes in one function affect another when combined. When finding the limit of a function product, you essentially look at how the product behaves near a particular point.

Function multiplication can be visualized as taking each pair of function outputs and multiplying them:
\[f(x) \cdot g(x)\]

This can be analyzed in different contexts like integrating products, dealing with functional combinations, or optimizing across multiple functions.
To simplify limits of function products, the product rule is used, which breaks down the process by evaluating each function separately first and then multiplying the results.

In our exercise, given that \( \lim_{x \rightarrow 4} f(x) = 16 \) and \( \lim_{x \rightarrow 4} g(x) = 8 \), multiplying these gives the result of 128 for the limit of their product, reinforcing the ease of using the product rule to simplify otherwise potentially complex function behaviors.