Finding the limit of f x consider it is difficult to

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Unformatted text preview: gain, transforming the original function will help us out: 2 x + 5 ( 2 x + 2) + 3 x +1 3 3 f ( x) = = =2 + =2+ x +1 x +1 x +1 x +1 x +1 3 lim 2 + =2 x→ +∞ x +1 17 Continuous Functions Definition A function y = f ( x ) is said to be continuous at point x0 if it satisfies the following requirements: satisfies y 1) x0 is in the domain of the function 1) x0 = 0 1 y= x is not in the domain of 1 x 2) There exists a limit 2) 0, x = 0 f ( x) = 1, x ≠ 0 0 lim f ( x ) = f ( x0 ) x → x0 lim f ( x ) exists, but it’s not x→ 0 equal to f ( 0) x y 1 0 x 18 Continuity and Smoothness Definition: A function is said to be continuous in set S if it is continuous at each point of S. continuous Continuity in a particular set has little to do with smoothness in that set. This function is continuous, This is but it is not smooth (so we know it won’t have limit values defined at its kinky points). points). It turns out that if we can differentiate a function in a particular set, it will It also be smooth in that set. in 19 Differentiable Functions Definition: a function is differentiable at point x0 if it has a lim derivative at that point, i.e. if the limit ∆x →0 f ( x0 + ∆x ) − f ( x0 ) exists. ∆x Definition: a function is differentiable in set S if it is differentiable at every point of S. every Definition: a function is continuously differentiable in set S if its ( 1) derivative in that set is a continuous function (notation: f ∈ C , f ∈ C ′ ) ⇒ Continuity ⇐ / Differrentiability lim ∆x → 0 f ( x0 + ∆x ) − f ( x0 ) ≡ { ∆x ≠ 0} = f ( x0 + ∆x ) − f ( x0 ) ∆x ∆x ( ) lim f ( x0 + ∆x ) − f ( x0 ) = f ′( x0 ) lim ∆x = 0 ∆x →0 ⇓ ∆x →0 lim f ( x0 + ∆x ) = f ( x0 ) ∆x →0 20 Economic Functions The many types of economic functions typically used include 1)Profit functions 2)Cost functions 3)Production functions 4)Demand functions 5)Supply functions The overwhelming majority of the functions actually used in the The economic analysis are assumed to be differentiable everywhere in a specific domain (often all real numbers). That, of course, implies that these functions are also assumed to be continuous everywhere in the domain. in 21 Which Statement is True? 1. Continuity at a point implies 1. differentiability differentiability 2. Differentiability at a point does not require 2. this point to be in the function’s domain this 3. Differentiability at a point requires 3. continuity continuity 4. Continuity at a point allows for different 4. left- and right-side limits left22...
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