Product law for limits

Definition

The Product Law for Limits states that the limit of the product of two functions is equal to the product of their limits, provided that these limits exist. Mathematically, if $\lim_{{x \to c}} f(x) = L$ and $\lim_{{x \to c}} g(x) = M$, then $\lim_{{x \to c}} [f(x) \cdot g(x)] = L \cdot M$.

5 Must Know Facts For Your Next Test

1. The Product Law for Limits can only be applied if both individual limits exist.
2. If either limit does not exist or is infinite, the Product Law for Limits cannot be used directly.
3. This law is useful in simplifying complex limit problems where functions are multiplied together.
4. It can be extended to any finite number of functions; the product of their limits equals the limit of their product.
5. The proof of the Product Law for Limits relies on the epsilon-delta definition of a limit.

Review Questions

Related terms

Sum Law for Limits: The Sum Law for Limits states that $\lim_{{x \to c}} [f(x) + g(x)] = \lim_{{x \to c}} f(x) + \lim_{{x \to c}} g(x)$, provided both limits exist.

Quotient Law for Limits: $\lim_{{x \to c}} [f(x)/g(x)] = (\lim_{{x \to c}} f(x)) / (\lim_{{x \to c}} g(x))$, provided both limits exist and $\lim_{{x \o c}} g(x) \neq 0$.

Limit:

A fundamental concept in calculus representing the value that a function approaches as its input approaches some value.