Limit Laws
1) lim C= C x—>a 2) lim X = a x–>a3) lim [f(x)+ – % x g(x)]= limf(x)+ – % x limg(x) x—>a x–>a x–>a 4)lim X^n = a^n x–>a
Limit laws are a set of rules that govern the behavior of limits of functions as the independent variable approaches a certain value. These laws are the foundation of calculus and provide a systematic way to evaluate limits without having to rely on algebraic manipulation.
The basic limit laws are as follows:
1. The limit of a constant function is the constant itself. That is, if c is a constant, then lim x→a c = c.
2. The limit of the sum or difference of two functions is the sum or difference of their limits. That is, if f(x) and g(x) are functions, then lim x→a [f(x) ± g(x)] = lim x→a f(x) ± lim x→a g(x).
3. The limit of the product of two functions is the product of their limits. That is, if f(x) and g(x) are functions, then lim x→a f(x) g(x) = lim x→a f(x) · lim x→a g(x).
4. The limit of a quotient of two functions is the quotient of their limits, provided that the denominator does not approach zero. That is, if f(x) and g(x) are functions, g(a) ≠ 0, then lim x→a [f(x) / g(x)] = [lim x→a f(x)] / [lim x→a g(x)].
5. The limit of a power of a function is the power of its limit. That is, if f(x) is a function and n is a positive integer, then lim x→a [f(x)]n = [lim x→a f(x)]n.
6. The limit of a composite function is the composite of their limits. That is, if f(x) and g(x) are functions, then lim x→a f(g(x)) = lim u→g(a) f(u), where u = g(x).
Using these rules, we can evaluate limits of functions and determine their behavior as the independent variable approaches a certain value. It is important to note that these laws only apply when the limits of the individual functions exist.
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