The Product Rule for limits is a rule used when you want to find the limit of the product of two functions. It states that:

$\lim_{{x \to c}} [f(x) \cdot g(x)] = \lim_{{x \to c}} f(x) \cdot \lim_{{x \to c}} g(x)$

This means that if you are trying to find the limit of the product of two functions $f(x)$ and $g(x)$ as $x$ approaches some value $c$, you can find the limits of $f(x)$ and $g(x)$ separately and then multiply them.

Here's how it works in simple terms:

1. Find the limit of each function: You first find the limit of $f(x)$ as $x$ approaches $c$, and then find the limit of $g(x)$ as $x$ approaches $c$.
2. Multiply the results: Once you have both limits, you multiply them together to get the limit of the product.

Example:

Let’s say you have two functions, $f(x) = 3x$ and $g(x) = 2x^2$, and you want to find the limit of their product as $x$ approaches 2.

$\lim_{{x \to 2}} [3x \cdot 2x^2]$

Step 1: Find the limit of each function individually:

• $\lim_{{x \to 2}} 3x = 3(2) = 6$
• $\lim_{{x \to 2}} 2x^2 = 2(2^2) = 8$

Step 2: Multiply the results: $6 \cdot 8 = 48$

So, $\lim_{{x \to 2}} [3x \cdot 2x^2] = 48$.

When to use the Product Rule:

• Both limits exist: The Product Rule works when the limits of both functions exist as $x \to c$.
• No need to simplify expressions: If the product involves complicated expressions, you might want to simplify each function first before applying the Product Rule.

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Would you like more examples or further explanation on the Product Rule?

Here are 5 questions related to the Product Rule:

1. How is the Product Rule used with infinity in limits (e.g., $\lim_{{x \to \infty}} [f(x) \cdot g(x)]$)?
2. Can the Product Rule be applied to more than two functions?
3. What happens if one of the limits in the Product Rule is 0?
4. How do you apply the Product Rule when dealing with trigonometric functions?
5. What is the difference between the Product Rule and the Sum Rule for limits?

Tip: When using the Product Rule, always check if both individual limits exist before multiplying them!