CALCULUS Chapter 1 Limits Continuity
:
and
*
What is a limit ?
-
the Ultimate guide- the number a function gets closer and closer
-
to as the input
approaches a certain number
(as approaches a, f(x)
mf(x) approaches ()
* Types of Limits * How to Evaluate Limits
1 Steps to
findlimf(x)
limit
-
Two sided - approaches
from both sides
A Direct substitution
· If the two sides don't agree ,
+he limit = BNE try to evaluate the function directly
- one-sided limits
f(a)
·
LEFT :
jima-f(x)
V r V
RIGHT
:ima f(x)
·
+
f(a) b =
f(a) = b f(a) =
2
where b = C Whereb is a real number
* What is a Discontinuity? B
. Asymptote . Limit
C found i .
Indeterminant form
[probably) [probably) example :
-
point where
a a function is not example : example :
2
x -
x -
2
continuous
im (3 x
I
-
lim = , -
2x - 3
x-> 13 1
be continuous at
-
=> to x = a :
Inspect with a graph
S
1) f(a) is defined or table to learn more
limf(c) exists failure function
2) x -> a
= about the
&
discontinuity at x = a
3) limf(x) =
f (a)
>a
x-
Try rewriting the limit in an
equivilant form .
V X ⑤
* Types of Discontinuities
Factoring Conjugates
#
E
. . G .
Trig identities
example : example : example :
Ya y
j x2 2
him-
sin(x)
O
f(x)
-
x -
f(x)
&
lim
x+ -
1x2 -
2x - 3
in o sin (2)
j
d x a e :
can be reduced to : can be rewritten as : can be rewritten as :
jump
removableality infiniteainuity
2
lim
discontinuity
x -
I
lim- lim
(a) (b) (a) x+ -
1x -
3 x+ 4x + 2 ccto2cos[x]
* by factoring
usingConjugatis using a
and cancelling tria identity
Shortcut : L'Hpital Rule
· Take the derivative of the I
A
numerator and denominator if
Toy evaluating the limit in
in an indeterminant form :
its new form !
an
H. Approximation
< Imlimes When
and
all
tables
else
can
fails ,
graphs
help approximate
=> then recalculate the new the limit .
limit !
* Shortcut for Rational Functions * Limits at Infinity/Infinite Limits
Let degree of numerator
n= ,
m= degree of 1) Infinite Limits (Vertical Asymtope]
denominator : ·
When f(x) -00 o r- o r , as e st a
= 0 CHA :
· If n < m , lim y = 0)
· If n= m ,
lim = ratio of leading coefficients
ex
: = ot tells you there a A a
· If n > m , lim = 00 or -oo (no HA)
2) Limits at Infinity (Horizontal Asymtopes)
·
W h e n -oo or- oo , look at the end behaviour
3T = 3
S
:lim
ex +
tells you theres a HA at y =
:
and
*
What is a limit ?
-
the Ultimate guide- the number a function gets closer and closer
-
to as the input
approaches a certain number
(as approaches a, f(x)
mf(x) approaches ()
* Types of Limits * How to Evaluate Limits
1 Steps to
findlimf(x)
limit
-
Two sided - approaches
from both sides
A Direct substitution
· If the two sides don't agree ,
+he limit = BNE try to evaluate the function directly
- one-sided limits
f(a)
·
LEFT :
jima-f(x)
V r V
RIGHT
:ima f(x)
·
+
f(a) b =
f(a) = b f(a) =
2
where b = C Whereb is a real number
* What is a Discontinuity? B
. Asymptote . Limit
C found i .
Indeterminant form
[probably) [probably) example :
-
point where
a a function is not example : example :
2
x -
x -
2
continuous
im (3 x
I
-
lim = , -
2x - 3
x-> 13 1
be continuous at
-
=> to x = a :
Inspect with a graph
S
1) f(a) is defined or table to learn more
limf(c) exists failure function
2) x -> a
= about the
&
discontinuity at x = a
3) limf(x) =
f (a)
>a
x-
Try rewriting the limit in an
equivilant form .
V X ⑤
* Types of Discontinuities
Factoring Conjugates
#
E
. . G .
Trig identities
example : example : example :
Ya y
j x2 2
him-
sin(x)
O
f(x)
-
x -
f(x)
&
lim
x+ -
1x2 -
2x - 3
in o sin (2)
j
d x a e :
can be reduced to : can be rewritten as : can be rewritten as :
jump
removableality infiniteainuity
2
lim
discontinuity
x -
I
lim- lim
(a) (b) (a) x+ -
1x -
3 x+ 4x + 2 ccto2cos[x]
* by factoring
usingConjugatis using a
and cancelling tria identity
Shortcut : L'Hpital Rule
· Take the derivative of the I
A
numerator and denominator if
Toy evaluating the limit in
in an indeterminant form :
its new form !
an
H. Approximation
< Imlimes When
and
all
tables
else
can
fails ,
graphs
help approximate
=> then recalculate the new the limit .
limit !
* Shortcut for Rational Functions * Limits at Infinity/Infinite Limits
Let degree of numerator
n= ,
m= degree of 1) Infinite Limits (Vertical Asymtope]
denominator : ·
When f(x) -00 o r- o r , as e st a
= 0 CHA :
· If n < m , lim y = 0)
· If n= m ,
lim = ratio of leading coefficients
ex
: = ot tells you there a A a
· If n > m , lim = 00 or -oo (no HA)
2) Limits at Infinity (Horizontal Asymtopes)
·
W h e n -oo or- oo , look at the end behaviour
3T = 3
S
:lim
ex +
tells you theres a HA at y =
