basic mathematics for class 12 (jee mains)

basic maths for class 12th

1. Page 1
Limits
Sat Apr 20 2024

Introduction to Limits Nedantu

2. Sage 2

Limits, Continuity and Differentiability

Lim —* —_

0 Buse

3. “page 3

Limits, Continuity and Differentiability

| Rationalization Method | Method

- _
x0 EE ree 7

If we put x = 0 here; it is indeterminate form
——t ___[N5-x+V5+x

xu V5 —x— V5+x Is ニュ + Ys

Lim *O5—x+V54x ]

0 (5-x) - (54x)

Lim ァ [Ns ー ェ +YVSTr )

x0

-2%
Le er) -

4. ‘page 4

Infinity limits and Some Example Nedanti,
General form of infinity limits
Lim

x>0

Lim
x>0

Lim
x>0

5. Page 5

Infinity limits and Some Example \edantu,

Ifm<n 0

Ifm > n and ayb, > 0 oo

If m>n and ayb, <0 —o

6. Page 6

Infinity limits and Some Example Nedantu,

"Example | [Example |

Lim 4+20+3 Lim _Xtl_
x>0 3-1 DO X+x+1

dim RAF E
N 220 Er q Ex)

7. ‘page 7

Infinity limits and Some Example Vedantin

amp
— | Example |
Lim 412043
>. x3_1 Lim x+1

— coo。 ol
fe)
im

rs Fein Lim «|: +1)

Lim xx
x>0
e Lim 1_
me Y

~ u

8. Page 8

> Limits, Continuity and Differentiability Nedantu

0 en es ー ax -b) = 수 , の の
- >00 HI
(a) az1 ,b=% JD así, b=-4
(> a=z , b=-3 (A) a=2,b=3
Tee (2012)
la Kant ~ axcerd EC?) -4 (ck vad 5
ズ っ の "ーーーーーーーーーーーーーーーー ーー ーー 1
K+]
(im
ズ っ oo

9. Page 9

Trigonometric and Exponential Limits Wed

Trigonometric Limits | ee
sinx is below

sin x =

Lim =1
x70 x © y=tanx Pr

. y=x tanx is above x
Lim sm 人 9 _ 1- A
fix) 0 fx |

y=sinx

Lim tanx + _

ーーーーー = x ax - 4
x>0 x = fe) Su de
Lim fanflx)_ + .
Fe Te mad.

Note: In all these standard limits , x is in radians

10. ‘page 10

Trigonometric and Exponential Limits Wed

Standard Limits of Inverse Trigonometric Functions

Lim == =1

x>0 x
Lim Sin" fl) -1
7 っ 0 fx)

Lim tam!x

x>0 x =]

Lim LAO,
700 700

11. Page 11

RE 1 [0192179 zen
Trigonometric and Exponential Limits

Example:

Lim [= sini 이 Where | x | is greatest integer function

x70
tin [7] =0
Rs D ン
Lim [x] = bom [ 1*] A
x>ol * as

Ve

12. Page 12

Trigonometric and Exponential Limits \edantı,

1- Cs2e= Lem:

fs Lim (1- cosx (2)

x>0 sie ¡NAS Ss の

La (Et). y
Mo Y ~

13. ‘page 13

Trigonometric and Exponential Limits Vedanti,

Example: |

Lim 1-cosx

x>0 x
Lim 2sin?(x/2) Remember this limit!!
x>0 x?

Lim ¿Sin (2) a
x90 NG の 4 4

1

14. “tifnits, Continuity and Differentiability Wed

y
| SC) = FR
ey him ELECO Cob 2 (2-1) dd ft = la)
204 a,
+
(a) 6004 and it equals S2 © O
(6) euh and ot equals -VR 168 1 48.

© hots met crust because (1) + ©
fa? dots net el becouse, safe tard Limit u not equal to
AGE hand, mt.
LIT Pyeious Year (1998

+ Ru し
Js (x-N a. ¥2 ] weCx-)1 _
LA = de Health.
Er = hina ~ 一

ne" um | wn (|
LHL? Lire pe hoo

h
ちっ ゃ o = = & B _
L? um ash] _ un BD pios = 2
E à à

15. ‘Limits, Continuity and Differentiability coo = 4. Nedantu,

He ‘a
シーー lim Ta (57490. MEIN, ek。 E
o tan Im
oo
(a) 4 (hy 3 (ed 2 CNA
EE Main (2013)
ae
tan ( u

Re

—

red = 의 4 (2)

4 + se
ze

. km er @) = ant > の

no

16. Timits, Continuity and Differentiability Nedantu

ES lim som (nee) ig equal to : Sin (1-6) = sin O
R>0 >
ce) TE (> 1 で ぐう - で DT

+ TEE Cram) - 2014

>

Lin (TE) © sn oa) = SCH) = ©
Te 0

mo enn Gs)

L len Er (Ti) a ÉD = Sin (sis)

HPO = ADO =

a
eb
Wir [em( Tin =
O

wz% っ っ

17. Page 17

an Et

2 ~ ai て ee
テニ E

= = 2 x À

N =.

Um

Ro

18. Page 18 . my = a ma

Trigonometric and Exponential Limits Wed

Exponential limits

Standard results ン
(Lim Gu e 1)

0 ASMA x: ak
D Ina _
= E D x Ina
Lim e-1 ‘ xIna | ~ 一
fo->0 {00 =Ina

+0 er =Ina {a>0)

These are standard formats
of exponential limits

19. Page 19

dnt =O |
Logarithmic limits and Rationalization method Vedanti,

Logarithmic Limits
\
malin Gi
Standard results A, mr) | (ON
4

2 Lim In(1 +x)
x70 x

A
Lim Indi+f@))

9 っ 0 fae 7.

20. Page 20

Limits, Continuity and Differentiability Nedantu

Lim 21 (1+x-a) '
620 Lim 1049)
L x>0 3-1

~ \ / 했지
(=)-0 = ^ es)
A+ 0 e

CDs

= A.
FT

21. Page 21

Limits, Continuity and Differentiability Nedantu

Example |

Lim In(1+x-a)

。 jm d+
xoa Lim 一 -- 一
(x-a) x30 *ー1
Lim A(A+x-a) Lim d+ けり
x-a20 (x-a) ンー nu
Lim (nl +@-ol) _ im (in A +3)
ah ea! =. ㅜㅠ
ここ ===
(GAL
Nr

22. Page 22

Logarithmic limits and Rationalization method
LM hea (e na

tt
| oa bg

L xian

= nal ¿n>0

il

23. Page 23 “m
um ATE nor!

xa AA

Logarithmic limits and Rationalization method Nedantu,
MAT
Lim x+x+x-3 Ee} aad cd
x 1 x-1 の

ー る =-1ー|
kai (2 Ea _

24. Page 24

Logarithmic limits and Rationalization method

[Example

Lim AHH)
xol ャ ー1

Lim &-ND+@&-D+@-D

x>1 ャ ー1

Lim 오그 4 #21 x 1
x>! x-1 x-1 x-1

= 1+ 2x (1) +3x (19

bee 22)
RL yy

L tes er
mob RZ

= By

Um x _ (6 DICELED)
RL RL - (< D

= E

25. Page 25

LH RULE

26. Page 26

LH Rule

It says that if Lim E is such that | it is either in 一 or 一 format ; then
|

Lim fa) _ Lim だ が @⑤

x= x>a ㆍ
7" 2) ge) Can again use if still in

0 oo
Functions are to be differentiated On 09
individually; not as a whole

Use this method
only for verifying answer

27. Page 27

Example :

Lim X-—sinx
x70

28. Page 28

Example :
As it is again giving 一 form so we have

Lim エーsiny .
e to apply LH rule one more time

x70 x
LL (1 - cos)
Tin dx
EM 4 。| ツー
Lim 2 dx
x>0 d
ーー y
dx Lim sine _ ML
x30 6x ーー 6
Lim 1-cosx |
x #0 3x2 Lim 1-cosx

x 0 3x2

29. Page 29

Example :

Lim x?
x70

30. Page 30

Example :
Lim x?-tan?x
x>0 xt
BUS Lim (x—tanx) (x+tanx)
x70 - A ur
x
Lim (x-tanx) (x+tanx)
* っ 0 a * x
ame)
Lim dx (x+tanx)
x30 ds x u

dx

Break this limit in
two parts out of
which one part is
giving a finite value

If some part of limit
is giving finite value
then no need to

apply LH rule on it

31. Page 31

Lim Te MY) (x+tanx)
x30 Ly x
Lim (1-secx) (x+tanx)
x30 3x2 x u. Ws
Lim (-tanx) (x +tanx)
* っ 0 3x2 并 — an
x x)
Lim | 그 /tan?x x ‚tan
x>0|73 FF} * | 된 a =
-1 =)
= —(1+1)=|=
3 ( ) su

32. Page 32

NS
Format of 1” and Miscellaneous problems

Format of 1°

33. Page 33

\edartu

Format of 1” and Miscellaneous problems

Format of 1°
Cc?

Li = Li
m fe) =1 and MN g(x) > ©

の ) E ) ° =
Consider the limit es 0600) 9 Le Lio (4 17 à -
s of the form |

It is simplified by following expression

8%)

5

Lim
. xa NW) -D x g(x)
Lim (DO = e 7

34. Page 34

Format of 1” and Miscellaneous problems Nedantu

[00006 2 (2) © Ge + a
aS ga

Y
y da 一

IS

a

x — 00

ん =

35. ‘page 35

Format of 1° and Miscellaneous problems Nedantin

Lim 1+ 237 we

Lim 000) 400 = @

xa

Lim (Ax) -1)x 800

Given expression becomes

Lin a+ 之 -Dx

Lim x

x30 X
< -@

36. its, Continuity and Differentiability Lar Wedantu
AZ

a a ： 2 ん
Qi Lee pz ln (ir tenia) wer equal to :

(o> 2 it KB 4 CIA
JEE Man (2016)

Gy ras
0 4 = 1 /
ast

LgeP = ん
tr)
Y
HE Han A 1 e*-p
ee 980
ie €
Han VA tan VA
_ pee an we TH )
” €
Lan AY
Les = = hy
= € = e 、 =

37. £
⑳ 5
O 1

©»

Ou

in (( (1-cos?(3z) sin*(42) :
z>0 cos*(47) (log, (22+1))" )) is equal to .

38. 2023 Main 8 April | ©

と (1—cos*(3z) sin?(4z) =
en (( cos*(4z) ) Fa 3) ii

AS _
1-ce's 24 2 (as
lin (va Wh (wie) O ，

#0 (5) (49° 202 9 で の
CON)! 소

Dx

A A

(る 6 +a)

39. Solution
ma) (mer) .,
개 ES e) e “ED
_[1x9x1 1x 64 +
| 0) E 1x2)

=9x2=18

40. Page 40

mA,

の lim (VSzHi+V3z-1)"+(VSz+1-V3z-1)" 3
m っ ュー の
25700 (<+v2?-1) +(z-v2?-1)

© is equalto 9
© is equalto q
lc] does not exist

[o] is equal to 27

41. lim (v3r+1+V32-1)°+H{V3r
2700 ee 를 ;

42. Solution lim (¥32+1+v32—1)*+ (v32-+1—v32—1)° „3

zoo (2/21) +(2- 1271)

= lim zs x (rt) + (yard 53)
200 (나 1 +) +( 1 x) |

CaO 他
~ 640 。

43. Page 43

44. Page 44

JEE Main 27" Jan 2024 S-2

~ Y
3+asinz+Pßcosze+log, (1-x) 1

If lim = 5
| 1>0 3tan?z 3°
\ v
\ then 2a - B is equal to:

ch
hm | 3 + “mw + BA a + hl

146 Où
ES Ds

45. Page 45

Limits Using Expansion Series

A 23 28 -

sino =x— 3 +5 > VE
一 一 a Ed ㆍ

csz=1-7 +7 -:..3Vr

3 975
tang=2+ +47 +...5Ve

ef =ltet 4% 4...¡Ve

pn | 3

h(1+%)=x-%+2

Yo fo,

46. Page 46

. 3+asinz + Bcosæ + log, (1— x)
If lim 들

+ ) 20 3tan?r
3! then 2a - 0 is equal to: 2
+f 1-22
x +

NE

New, Cot 9 タ =o
, Gulf tum =o
fa Limi be be ruse ㆍ

47. Page 47

Aro >

ae

Lv A 一 file Y

mao

9
=

NO

48. Page 48

49. Page 49

JEE Main 27" Jan 2024 S-2

3+asinz+Pßcosze+log, (1—zx) 1

If lim = 5
f 230 3tan?r 3”
\ \ then 2a - 0 is equal to:

©"
OL

50. Page 50

JEE Main 31 Jan 2024 S-1

2\sin x

. € —2|sinz|—1
lim

\ = x”

( A ) is equal to 1
6 ) Does not exist

c ) 16 equal to 2

D ) Is equal to - 1

51. ‘Page 51

2\sinz| _ 2 [sin «| — 1

52. nn 52

Xo

~ に alsimx| + La » ㅣ

>

+ ere |
+ eine y | sive? Isa) (
一 ae

:Q

53. 2/ Sino}
= lim € ー 2 JSinx] —| , six
SO
| Sima] 5

54. 21Sinoe)
= Uw € - 2 Sina] I an
^
30 [Sox]? ar
26 -
= lim € -2t-) そ = /Sinx)
fo 一 テー
5 te 일으 의 ta ete) = O
E 270 RE

55. JEE Main 31 Jan 2024 S-1

2\sin x

. € —2|sinz|—1
lim

\ = x”

( A ) is equal to 1
6 ) Does not exist
CE) Is equal to 2

D ) Is equal to - 1

56. Sage 56

All the very best :-)

DREAM ON!