Mastering Limit Laws: Rules To Evaluate Limits Of Well-Behaved Math Functions

Limit laws

Sum lawlim ( x→ a) [ f(x) + g(x) ] = lim ( x→ a) f(x) + lim ( x→ a) g(x)Difference lawlim ( x→ a) [ f(x) – g(x) ] = lim ( x→ a) f(x) – lim ( x→ a) g(x)Constant multiple lawlim ( x→ a) [ cf(x) ] = c lim ( x→ a) f(x)Product lawlim ( x→ a) [ f(x)g(x) ] = (lim ( x→ a) f(x))(lim ( x→ a) g(x))Also, if one of the limits don’t exist, then the product of the two limits doesn’t exist.Quotient lawlim ( x→ a) f(x) / g(x) (lim ( x→ a) f(x)) / (lim ( x→ a) g(x))Power lawlim ( x→ a) f(x)^n = [lim ( x→ a) f(x)]^nRoot lawlim ( x→ a) rt^[(x)]. =. rt^[lim ( x→ a) f(x)](rt = root of a number but not necessarily the square root)Alsolim ( x→ a) x = a and. lim ( x→ a) c = cDirect substitution propertyif f is a polynomial or a rational function and a is in the domain of f, thenlim ( x→ a) f(x) = f(a). However, you need to make sure that the function is continuous at a. If not, then direct substitution won’t work. Also, side note for piecewise functions with limitsThe value of a limit as it APPROACHES a does not depend on the value AT a.

Limit laws are a set of rules that help us evaluate limits of functions. The following are some of the important limit laws:

1. Sum and Difference Law: The limit of the sum or difference of two functions is the sum or difference of their limits. Mathematically, if lim f(x) = L and lim g(x) = M, then lim (f(x) ± g(x)) = L ± M.

2. Product Law: The limit of the product of two functions is the product of their limits. Mathematically, if lim f(x) = L and lim g(x) = M, then lim f(x)g(x) = LM.

3. Quotient Law: The limit of the quotient of two functions is the quotient of their limits (provided the limit of the denominator is not zero). Mathematically, if lim f(x) = L and lim g(x) = M (where M is not zero), then lim (f(x)/g(x)) = L/M.

4. Power Law: The limit of a power function, where the exponent is a constant, is the power function of the limit of the base. Mathematically, if n is a positive integer and lim f(x) = L, then lim [f(x)]ⁿ = Lⁿ.

5. Exponential and Logarithmic Law: The limit of the exponential or logarithmic function is the exponential or logarithmic function of the limit. Mathematically, if lim f(x) = L, then lim e^f(x) = e^L and lim log_a(f(x)) = log_a(L), where a is a positive constant.

Using these laws, we can evaluate many limits algebraically without using any advanced methods. However, it is important to note that these laws apply only to the limits of well-behaved functions.

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