Class notes

Calculus 1: Limits

6 views 0 purchase

Course
Calculus 1

Institution
John Abbott College ( )

This document covers the basics of limits, asymptotes, methods of factorisation, finding a continuity condition, continuities, discontinuities, and piecewise functions. Not only does the document include clear, easy to understand detailed notes, but also a practice test at the end with the answers.

[Show more]

Preview 4 out of 31 pages

View example

Uploaded on November 27, 2021
Number of pages 31
Written in 2021/2022
Type Class notes
Professor(s) Michelle bokwood
Contains Limits

calculus
cal
calculus 1
math
math
limits
intro to calculus
clear
digital
ipad
digital notes
ipad pro
pdf
download
easy
continuity
discontinuity

CA$15.23

Added

Add to cart Add to wishlist

100% satisfaction guarantee
Immediately available after payment
Both online and in PDF
No strings attached

LIMITS
Limits : The value that a function approaches as the input

approaches some value

realityvsexpectation
✓
^

,
Expectation :
limfcx )
✗ → a
in undefined
' "
2

GABBY Reality
. ,

: FCX )

define,
, DNE : does not exist
-3 2 I 2 3
1
-
-

-1
Ex : in the
vicinity of ✗ = -2
,

-2 EXP Lim FCX)= 2
✗ → -2
since there is no •

-3 REAR f- C- 2) =D NET

More ex :

Lim fcx )= DNE him f- ( X ) -_ 2
✗ → I ✗ →2

f- (1) = I f- (2) = 0

✗ = -
I Limfcx) =-3 EXP
✗ → -
I
-

4 %••☒
-
f- C- 1) = -

2 REAL
.

-
2 %Aµ☒
✗= I limfcx ) =
-

I
- Bog ✗ → I

Fatone
2
-
-

,

✗= 2 Iim f- 1×7=0
q••☒ -
-2 ✗ → 2

-
f- (2) = 2
,

-
-4
✗ =3 Iim f- ( x )= DNE
✗ → 3
.

f- (3) = I

✗= 4 limfcx ) =3
✗ → 4

f- (4) = 4

, and
-

+ :
✗ at look at function
→
means to the coming
from the
right
✗ → a- means to look at the function
coming
from the left

one sided limits :
2 limits from left and the
coming the

for the variable ( jump )
right same

Two sided limits t limit
coming from left and the
:
the

for the
right same variable .

Always DNE

Ex :

}
Iim f- 1×1=2

.{i)fq*+ne+w
✗ → 1- one sided limits

.am?.. *zLimfCx
Iim FCX ) =3
-- 2 ✗ → It .

Lim FCX ) = DNE → two sided limits
☒ → I

f- (1) = -

I
. sided limits
,

) exists only if the
✗ →a
-2
one sided limits , lim f- ( X)

I and Lim Flx )
✗ → a-
exist
✗ → at

and

agree .
Iim = Iim =/ im
✗→ a ✗ → at ✗ → a-

1 Iim f- ( X ) = 2
✗ → it
-

man -2 Iim fix)= 3
* → -

zt
-

I ¥8 limfcx ) =3
✗ → of
-

Iim f- Cx) =3
- -2
✗ → 0

-
limflx)= DIVE
✗ → I

f- C- 2) = 2

, DISCONTINUITY
continuity : A function is continuous if limflx )=limflx_- flat
✗ → a- ✗ → at
and all exist

Discontinuity : A function is discontinuous at ✗ =a if exp =/ real

The function has a
discontinuity at X=a if we lift

at ✗ =a to draw the of the function
our
pen graph .

..¥
^

/
}
"" "

:÷÷÷sa+×=
✗ = 10
,
8- since exp = real and we don't
I f- ( 17=2
6- *a→
I
4-
Iim

}
FCX )= DNE
2- neat 1 ✗ → 5 fcx ) is discontinuous at
Iimfcx )= 4 EXP
,
✗ =5 since exp # real and
I 1 I 1 I > ✗ → 5-
we must lift our pen
4 8 f-(5) 6 REAL =
JUMP DISCONTINUITY
limflx )=6
✗ → 5-1
I

} }f¥fi§,ndiI%Ip¥¥a
limfcx )=5 fix ) discontinuous at Iim f- 1×1=+00 at
✗ → 8 =8 since
✗ exp =/ real ✗ → 10 ,

f- (8) = DNE REMOVABLE DISCONTINUITY f- ( lo ) = DNE INFINITE DISCONTINUITY

Types of discontinuity
) ¥ limflx ) but Can
Jump When
linga,ECx they both exist be equal or
: .

✗ → a-
not to f- (a)

none infinite
-

^ are
r n

-

can occur at :

BET breakpoints in piecewise
EA
functions ( 2 > × )

#
>
§
s
absolute values
giving
( ¥¥z )
.

removable :
when limfcx)= limftx ) =/ flat but limfcx ) exists

,,.••none
✗ → a- ✗ → a
exp =/ real s

'

IIMFCX ) #
-

^
same f- (a)
thing as
✗ → a

are man ,+e

s
-

can occur at :

breakpoints of piecewise functions
.

( x > 2) →
§

, Infinite : when theres an
asymptote ( HA or VA )
can occur at :

limflx ) → IN and/or Iim →
± a
§ ,
c. =/ 0
g- →
§ ,
c -1-0 breakpoints
Piecewise function
of a

✗ → a- ✗ → at

E. or
E. →
§
^ ^
^ 1 ^
1 1 1
1
I 1 Kainuu
l l
l
- l
l
µ%s8 l
l
s
l s
s l s

1 i I 1
Limits and discontinuities
limflx) -_ too f- 1- 1) =3
ex :
✗ → -
2-
'
I ^ Iimflx )= I
1
limflx )= -
N
✗ → O
-4
l ✗ → -2T

/ wñ#i - ME
limflx )= ☐ NE
HO )= /

I -2 -
-
- - - -
-
- -
- - - -
✗ → -2
limflx )= -

l
l ✗ → 2-
f- (2) = DNE
-

I
limflx )= -
I
Iimflx )=o ✗ → 2T
I ✗ → -
l
-

l -

l
tim f- ( ✗ 1=3 limflx )= -

l
l ✗ → 2
-
-2 ✗ → -
it

| Iim f- 1×1=0
-

limflx )= DNE ✗ → -00
I ✗ → -
l

-4
Iimfcx ) 2
-

l
1 f- (2) =3 __

✗→ oo

Find the discontinuities and their
type
:

11=-2 : infinite

✗ = -1 :
jump
11=2 : removable

DIRECT SUBSTITUTION
if f- ( X ) is continuous at X=a
,
then limfcx)= f- (a) .
To evaluate the
✗ → a

limit , plug in a .
If it
gives you a number like 2 orbs ,
that's
your
answer .
If
you get
8- ,
facto rise 14 methods ) see next pages .
If
you get § ,

it's either
-

do ,
so or DNE .

Evaluate the limits :

tim .
✗2+4=8
→
22+4 Iim ✗ + 4- =3
✗ → 5
✗ → 2

Iim ✗ +3=6 lim X -
3=0
✗ → 3×2-4 5 ✗ →
3×2-9 ☐

Iim ✗ +3 =I
2
✗
X2 -4
→
0

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller jessdunberry. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for CA$15.23. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

75632 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling