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
- The Constant Multiple Law can be applied to both finite and infinite limits.
- This law simplifies the calculation of limits when a constant factor is involved.
- It is one of several limit laws that facilitate easier computation of limits in calculus.
- Understanding this law helps in solving more complex limit problems by breaking them into simpler parts.
- 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?
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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: $\lim_{{x \to c}} [f(x) \cdot g(x)] = (\lim_{{x \to c}} f(x)) \cdot (\lim_{{x \to c}} g(x)$
