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UntitledLimits and Continuity
Will both cars will get to the restaurant if they take their respective roads???
Limits:
"the limit as x approaches c equals N"
c
L
x
ε
δ
f(x)
f(x) has a limit L as x approaches c if: for any arbitrarily small , there exists a correpsonding .
In other words: given any x ≠ c, we can find a new value of x that will make f(x) closer to L
ε
Lim f(x) x→c
Lim f(x) x→c+
= Lim f(x) x→c
c LH limit RH limit Limit
1
2
4
2
0
1
2
4
We only need one of the handed limits to exist at an endpoint when
the domain is restricted!
Lim f(x) for the following values of c x→c
Find
WhatisLimtan(x)?Justifyyouranswerwithasketch. x π/2
Limits from a numerical approach
x f(x)
1. You can apply • substitution • factoring/simplifying • conjugates
2. The limit of the sum of 2 functions is the sum of their limits
a) same for difference, product and quotient
limit on bottom not zero!
Example #1: Example #2:
Example #3: Example #4:
a) b)
Example #5:
Example #6:
Unit 1 Limits 1. f(x) approaches L as x approaches c.
2. f(x) has a limit L as x approaches c if, for any , there exists a such that
how close f(x) is to the limithow close x is to c
You can always find an x which will make f(x) as close as you want to L (but not necessarily equal to L !)
Limits of Piecewise Functions Algebraically
Find Lim y(x), given x→1
Check the handed limits when taking limit at boundary!!!
P ul l
P ul l
Menu 4 4
1.
must know limit !
shortcut 2 if the denominator is of higher order, the limit equals zero.
Practice
3. If or , then y = b is a horizontal asymptote.
4. Vertical asymptotes occur where the limit = +/
vertical asymptote: x = 3
1. Find the equation of the horizontal asymptote of f(x) =
2. Find the equation of the vertical asymptotes of f(x) =
3. Find the value of
5. End Behavior Model (EBM) function that approximates another function as
ex) f(x) = x2 4x EBM is g(x) = x2
dominant term as
Find and EBM for f(x) = x4 + 4
x3
Then
then
Will both cars will get to the restaurant if they take their respective roads???
Limits:
"the limit as x approaches c equals N"
c
L
x
ε
δ
f(x)
f(x) has a limit L as x approaches c if: for any arbitrarily small , there exists a correpsonding .
In other words: given any x ≠ c, we can find a new value of x that will make f(x) closer to L
ε
Lim f(x) x→c
Lim f(x) x→c+
= Lim f(x) x→c
c LH limit RH limit Limit
1
2
4
2
0
1
2
4
We only need one of the handed limits to exist at an endpoint when
the domain is restricted!
Lim f(x) for the following values of c x→c
Find
WhatisLimtan(x)?Justifyyouranswerwithasketch. x π/2
Limits from a numerical approach
x f(x)
1. You can apply • substitution • factoring/simplifying • conjugates
2. The limit of the sum of 2 functions is the sum of their limits
a) same for difference, product and quotient
limit on bottom not zero!
Example #1: Example #2:
Example #3: Example #4:
a) b)
Example #5:
Example #6:
Unit 1 Limits 1. f(x) approaches L as x approaches c.
2. f(x) has a limit L as x approaches c if, for any , there exists a such that
how close f(x) is to the limithow close x is to c
You can always find an x which will make f(x) as close as you want to L (but not necessarily equal to L !)
Limits of Piecewise Functions Algebraically
Find Lim y(x), given x→1
Check the handed limits when taking limit at boundary!!!
P ul l
P ul l
Menu 4 4
1.
must know limit !
shortcut 2 if the denominator is of higher order, the limit equals zero.
Practice
3. If or , then y = b is a horizontal asymptote.
4. Vertical asymptotes occur where the limit = +/
vertical asymptote: x = 3
1. Find the equation of the horizontal asymptote of f(x) =
2. Find the equation of the vertical asymptotes of f(x) =
3. Find the value of
5. End Behavior Model (EBM) function that approximates another function as
ex) f(x) = x2 4x EBM is g(x) = x2
dominant term as
Find and EBM for f(x) = x4 + 4
x3
Then
then