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”
(a,f(x)) shows ( )=L
F(n) NOTE: L≠f(x)
Let’s consider ( ). As t→2, there is no one single real number L such that S(t)→L.
∴ ( ) DNE , if x≠0
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
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
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