Constant multiple law for limits

Definition

The Constant Multiple Law for limits states that the limit of a constant multiplied by a function is equal to the constant multiplied by the limit of the function. Mathematically, if $\lim_{{x \to c}} f(x) = L$, then $\lim_{{x \to c}} [k \cdot f(x)] = k \cdot L$ where $k$ is a constant.

5 Must Know Facts For Your Next Test

1. The Constant Multiple Law can be applied to both finite and infinite limits.
2. This law simplifies the calculation of limits when a constant factor is involved.
3. It is one of several limit laws that facilitate easier computation of limits in calculus.
4. Understanding this law helps in solving more complex limit problems by breaking them into simpler parts.
5. It applies to both continuous and discontinuous functions as long as the basic limit exists.

Review Questions

• What does the Constant Multiple Law for limits state?
• How would you apply the Constant Multiple Law to find $\lim_{{x \to 2}} [5 \cdot f(x)]$ if $\lim_{{x \to 2}} f(x) = 3$?
• Does the Constant Multiple Law apply to infinite limits?

Related terms

Limit: The value that a function approaches as the input approaches some value.

Sum Law for Limits: The limit of a sum is equal to the sum of the limits: $\lim_{{x \to c}} [f(x) + g(x)] = \lim_{{x \to c}} f(x) + \lim_{{x \to c}} g(x)$.

Product Law for Limits: $\lim_{{x \to c}} [f(x) \cdot g(x)] = (\lim_{{x \to c}} f(x)) \cdot (\lim_{{x \to c}} g(x)$