Limit laws意思
"Limit laws" is a term used in mathematics, specifically in the field of calculus, to describe certain properties or rules that govern the behavior of limits. Limits are a fundamental concept in calculus, where they are used to define continuity, derivatives, and integrals.
Limit laws are rules that help in evaluating limits of functions in a systematic way. Some of the common limit laws include:
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Constant Multiple Rule: If the limit of a function (f(x)) as (x) approaches a certain value (say (c)) exists and is equal to a constant (k), then the limit of (k \cdot f(x)) as (x) approaches (c) is also (k) times the limit of (f(x)). In other words, if (\lim{x \to c} f(x) = L), then (\lim{x \to c} k \cdot f(x) = k \cdot L).
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Sum Rule: If the limit of a function (f(x)) as (x) approaches a certain value (say (c)) exists and is equal to a certain value (L), and the limit of another function (g(x)) as (x) approaches (c) exists and is equal to a value (M), then the limit of the sum of these two functions, (f(x) + g(x)), as (x) approaches (c) is (L + M). In other words, if (\lim{x \to c} f(x) = L) and (\lim{x \to c} g(x) = M), then (\lim_{x \to c} [f(x) + g(x)] = L + M).
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Product Rule: If the limit of a function (f(x)) as (x) approaches a certain value (say (c)) exists and is equal to a certain value (L), and the limit of another function (g(x)) as (x) approaches (c) exists and is equal to a value (M), then the limit of the product of these two functions, (f(x) \cdot g(x)), as (x) approaches (c) is (L \cdot M). In other words, 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).
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Quotient Rule: If the limit of a function (f(x)) as (x) approaches a certain value (say (c)) exists and is not equal to zero, and the limit of another function (g(x)) as (x) approaches (c) exists and is not equal to zero, and the limit of (g(x)) divided by (f(x)) as (x) approaches (c) is equal to zero, then the limit of the quotient (f(x)/g(x)) as (x) approaches (c) is equal to (L/M), where (L) is the limit of (f(x)) and (M) is the limit of (g(x)). In other words, if (\lim{x \to c} f(x) = L), (\lim{x \to c} g(x) = M), and (\lim{x \to c} [f(x)/g(x)] = 0), then (\lim{x \to c} [f(x)/g(x)] = L/M).
These limit laws are fundamental in evaluating limits of functions, and they are often used in conjunction with other techniques such as direct substitution, factoring, and canceling to find the limit of a function.