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limits[ch10.1-10.3](sept25).docx

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Department
Mathematics
Course
MATA32H3
Professor
Raymond Grinnell
Semester
Fall

Description
Limits (§10.1-10.3) The idea or concept of limit is the foundation of all of calculus (i.e. derivatives and integrals). Main definition (p.461) (limit of a function at one point). Let y=f(x) be a given function and a∈R We write ( ) where L∈R to mean that the function values f(x) can be made arbitrarily close to L (and they stay close to L) provided X is close to a) but x≠a. 1| ( ) |=very small 2| |=very small, but +0 If there is no real number L as above, then we write ( ) DNE (Does Not Exist). Diagrams for “limit situations” Y F(a) (a,f(x)) shows ( )=L F(n) NOTE: L≠f(x) X S t(time) Let’s consider ( ). As t→2, there is no one single real number L such that S(t)→L. ∴ ( ) DNE , if x≠0 Let g(x)= 32, if x=0 g(32)= , g(0)=32 The graph shows ( ) DNE because L ∈ R ∋: g(x) →L as x→0, x≠0. A lot of our interest is being able to calculate limits. “There exists” ∃, there does not exist , “such that” Tools for finding limits- i.e. limit properties (pg. 463-465) Polynomial Property (pg.464): if y=p(x) is a polynomial, then ∀a∈R ( )=p(a) (i.e. obtain limit by plug-in) (∀= “for all”) ( ) Rational function property: Let r(x)= rational function and a∈R. Write r(x)= ( ) where p and q are polynomial. ( ) If q(a)≠0, then ( )= r(x) = ( ) ( ) ( ) = ( ) = ( )= r(a) (see the lists of limit properties p 463-465) We are concerned with the question find ( ) where f(x) is a given function and a∈R given, or determine that it DNE. Now we look at what’s called a “ – form” of a limit. EXAMPLE: r(x) = is a rational function p(1)=0 both are 0 so we call it a 0/0 form q(1)=0 To evaluate manipulate algebraically. ( )( ) = ( )( ) ( ) = ( ) = means x is close to 1 but x≠1 The limit means that gets close to as x gets close to 1. EXAMPLE2: Let g(x) = √ , find * ( ) ( + Solution: we have a “ form”. √ √ = (√ √ ) √ √ ) = [ √ √ )] = * + ( )(√ √ = *√ √ + = = √ √ EXAMPLE3: This example shows how a derivative calculated from “1 principle” is actua
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