Calculus Notes

, lesson 1 :( Part 4) One-sided limits
-




let tail be a
function defined at every number in some open
interval tail . We ray that the limit of fat as ✗ approaches a
from left is L .
denoted
let lim 1-1×1 .
L
2+4×1-3

{ ifthexaluevotflxgetdoreranddosertol.F
, ✗→ at
✗ if ✗ ± .




*
flx) = -1
if -14<2


/ it ✗ 22
as the values of ✗ get closer and closer to a



butlessthana.to/ulion
approaches ✗ to -1


Example 4×+7 ✗ a




:{ g-
it
-




} |
✗ THI -
Mt 4×1-3 ✗ FIX)=Ñ -1 Determine lim.lk) ft) ✗
A it -21×4
-1.5 -0.75 .gg -
0.75 ✗ → -2 it ✗ It
-1.17 -0.3111 -0-8 -0.36 -21 endpoint internals )
solution :
since ✗→
of
-1.003 -0.005991 -0.99 -0.0199
will one-sided limits
we use to evaluate linn fix)
µ
y , , , ,, ,
,
, ,,, , , ggggg .

g. gggg ,


,
,,, ,gggg

Iim flx) :
tim (4×+7) = 41-271-7 ⇐ -1

to
-




-2
approaches 2 →
✗ I
Solution ✗ ✗ →
lim fan DNE

bright) tim 1-212 4
✗→ -2
( left ) tim
: ±
:




"" ×Ñ



{ }
"" " "" "
✗ """ " " ✗ "




1.9 d. I

1.99 2.01 Iim tu
✗→ I
1- 999 g. 001
g. gggg
gy , , , gym µ , , ,µµ,n, ,
, myna, , , ,µµ , ,
,
,, , ,n , .gg , ,m , ,



suppose tim 1-1×1=0 f approaches 0 through positive Iim the ✗ 2=14--1
.




If values , Iim
✗→ a ✗ → 1- ✗ → 1-
Iim fix) DNE
Ot * 1
flx)
YY f-
write im
9-
→ '
we -


-
a
: =


✗ → It , ,



similarly , it t approaches 0
through negative tattles , we

Write fly) →
O
-


N Iim HD :
Iim Fita =¥t= TO
; Iim fix) DNE
-
-


✗ → -2
-




✗→ -
y X -7-2


Remark :
1in fix) : Iim 5×+2=1-21%12 = To :O

at at
Ot
-



✗→ ✗→
It fly) and calm then
-




→
as ✗→ a ,
n is ,



""
"" " " ""
1149
let flx) =Ñ✗ .
Eval . Iim to)
✗ →•
If fall -70 and
-




as ✗ →
as n is Alan ,
then
IIM Ht) B-✗ RI To
TFF) Iim
=
tim
DALE
=
.



✗→ a mg ✗ →3



Iim flx) Iim B-3-1 -_3-3-Ñ= No -



Iim tlx ) DNE




}¥¥#
:
;
✗→ 3T ✗ -73 ✗→ gt
Example
let TIM :# .
Evaluate 1in flx) im Ht ) :


iimf-J-z-s-T-rt-ojiimfc.no
glim
✗ → 3- ✗→ 3 ✗ →
✗ -7 q
-




Fata -1K ) = lim Kita =
= To
x v
✗→ a
one-sided limits
-




use