Limit & Continuity (Solved MCQs and Notes)

Notes 85 Pages

Joint Entrance Examination (Main)

Mathematics

Contributed by

Nikita Narasimhan

315
Unit-8
Limit and Continuity
Important Points
Real Value Function:
If function f : A

B (Where AA

R and B

R)
is defined then f is a real function of a real value.
Some useful real functions of real value.
(1) Constant function :
 
Ax,cxf,BA:f 
(where
RA 
,
RB 
,
 B,A
) is called a constant
function.
Here c is fixed element of c.
N.B,d
 
f
R c
(2) Identity function:
 
Ax,xxI,AA:I
AA

is called identity function,
N.B. : Indentity function in one-one and onto.
(3) Modulus function:


0URR:f


 






0x,x
0x,x
|x|xf
is called modulus function.
(4) Integer part function Floor function.
(A)
zR:f 
, f(x) = x the largest integer not exeeding x’ is called integer part function.
It is denoted by [x] or [x].
 
   






2777f,41.31.31.3f 
(B) Celling funtion :
 
xxf,zR:f 
the smallest integer not less than x, is called celling funtion and
denoted by [x].
 
3.1 3.1 2, 7 3f
 
     
 
 
 
(5) Exponential Function



 Ra,axf,RR:f
x
is called exponential function.
N.B. : 1. For a = 1, R
f
= {1} i.e, f is constant function.
2. for 0 < a < 1 then this function is decreasing function.
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316
3. For a> 1 then this function is an increasing function.
4. Graph of any exponential function always passes through the point (0, 1)
(6) Logarithmic function :


 
: , log , 1
a
f R R f x x a R
 
   
,is logarithmic function.
N.B. :
1. Logarithmic function is an inverse function of an exponential function
Also exponential function (with R
+
-{1} is an inverse function of logarithmic function.
i.e. Both are the inverse function of each other.
2. Working rules of :
(A) Exponential Function (B) Logarithmic function
For


Ry,x,1Rb,a 

For



 Ry,x,1Rb,a


yxyx
aa,ai






ylogxlogxylogi
aa

   
xx
x
baabii 
 
ylogxlog
y
x
logii
aaa









 
 
xy
y
x
aaiii 
 
Rn,xlognxlogiii
a
n
a

 
x
x
x
b
a
b
a
iv 






 
alog
xlog
xlogiv
b
b
a

 
yx
y
x
a
a
a
v




01logv
a

 
x
x
a
a
1
vi




1alogvi
a



Rxxaxlogavii
a

(7) Polynomial function :
 
;a...........xaxaxaxf,RA:f
0
2n
2n
1n
1n
n
n





 
0a,n...,.........3,2,1,0i,Ra
n

is called a Polynomial function
  
0Nn,A,RA 
(8) Rational function :
 


 
xq
xp
xf,RA:f 
where p(x) and q(x) are polynomial function over (A on) and
 
,Ax,0xq 
is called rational function.
Page 2
317
(8) Signum Funtion:
 
1,0,1R:f 
 









0x,1
0x,0
0x,1
xf
is called signum function
N.B. :
 








0x,0
0x,
x
|x|
xf
(10) Trigonometric functions : Inverese Trigonometric functions :
Function Domian Range Function Domian Range
Sine R [-1, 1] Sin
-1
[-1, 1]








2
,
2
Cosine R [-1, 1] Cos-1 [-1, 1]
 
,0
Tangent
 














zk
2
1k2
R
R tan-1 R








2
,
2
Co-tangent









zk
k
R
R Cot-1 R
 
,0
Secant
 













zk
2
1k2R
R-(-1,1) Sec-1 R-(-1, 1)
 








2
,0
Cosecant









zk
k
R
R-(-1,1) Cosec-1 R-(-1, 1)
 
0
2
,
2









(11) Even function :
If
 
 A,RA,RA:f
is a function
AxAx 
yLku
   
,Ax,xfxf 
then f is called an even function.
(are all even functions define on their respective domain set
 
0zn 
)
(12) Odd function :
and
 
 A,RA,RR:f
and
AxAx 
yLku
   
,Ax,xfxf 
then f is
Page 3
318
called an odd function
and odd function define on their respective domain

Limit of a fuction :
Let f(x) be a function define on a domain comtaining some interval but may be in the
domain of f If for every f

> 0 there exists some

> 0 there exists
 
  xfDx,ax,axa
f
whenever
a
x 
we say left limit of
f(x) is l or
 


xf
ax
lim
.

Right limit of a fuction :
If f(x) is function defined in some interval
   
0h,a,ha 
and for every

> 0, there
exists

> 0 such that
   
a,ax,xf  
then we say tight limit of (x) is 1 as
x

a
+
OR
 


xf
ax
lim

Algebra of Limits :
Let
 
lim
f x
x a
exist and be equal to
 
xg
ax
lim

exixt and be equal to,
Then (1)
    
xgxf
ax
lim


exist and
        
mxg
ax
lim
xf
ax
lim
xgxf
ax
lim







(2)
    
xgxf
ax
lim

exist and
        
mxg
ax
lim
xf
ax
lim
xgxf
ax
lim

















(3) If
0m 
íkku


 
xg
xf
ax
lim

exist and
 
 
 
 
m
xg
ax
lim
xf
ax
lim
xg
xf
ax
lim






If
 
cxf 
(C = constant) in a constant function then
 
cc
ax
lim
orcxf
ax
lim




Page 4
319
Theorem : 1
Nn,ax
ax
lim
nn


Theorem : 2 If individual
n.......,.........3,2,1
then
 
x;f
ax
lim

   














n
1i
ax
n
1i
x;f
ax
lim
x;f
ax
lim
Limt of a polynomial :
If f(x) = c
n
x
n
+ c
n-1
x
n-1
+......+ c
0
, x

0, c
i


r i =0, 1, 2, ........n) g(x) is a polynomial of
degree then
 
 
 
 
 
 
 
lim
f x
lim f x f a
x a
h a
lim
x a g x g a
g x
x a

  


N.B. :
1. In a rational funtion
 


 
 
0xg,
xg
xf
xh 
f(x) and g(x) have a same factor (s)
   
Nkax
k

with same index K then after cancellation of the factor (s) (x-a)
k
, we have the
limit by substituting x = a in the remaining part of the rational function.
 


 




   
xqax
xpax
ax
lim
xg
xf
ax
lim
xh
ax
lim
m
k







 
 
 
mk,
xq
xp
ax
ax
lim
mk




= 0, If k -m

N


 
aq
ap

if k = m
= Limit does not exist, If k < m
2.
  
Rax,axNn,a.n
ax
ax
ax
lim
1n
nn





N.B. : This result is true even for n

R, while
ax,Ra,Rx 

Rule of substitution (OR) Rule of Limit of a composite Function :
Page 5
320
Suppose
 
xf
ax
lim

exist and
 
bxf
ax
lim


and
 
yg
by
lim

exist and
 


yg
by
lim
then
  


xfg
ax
lim
Two important rules :
1. If f(x) < g(x) in the same domain and both
 
xf
ax
lim

and
 
xg
ax
lim

exist, then
   
xg
ax
lim
xf
ax
lim



2. If
     
21
fff11
DDDx,xfxfxf 
and if
 
xf
ax
lim
1

and
 
xf
ax
lim
2

exist
and are
   




xf
ax
lim
xf
ax
lim
21
exists and is equal to l.
Some important results of trignometric functions, limits.
1.
2
|x|0,x,1
x
xsin
xcos


(2)
Rx|,x||xsin| 
(3)
Rx,1xcos
2
x
1
2

Limits:
(1)
0|x|
ax
lim


then
 
0|xf|
0x
lim



 
0xf
0x
lim


(2)
0xsin
ax
lim


(3)
1xcos
ax
lim


(4)
asinxsin
ax
lim


and
 
Ra.acosxcos
ax
lim


(5)
1
x
xsin
0x
lim


Page 6
321
(6)
1
x
xtan
0x
lim



x
and

x
and infinite limit
(1)
 
:xf
x
lim

If for every
,0
there exist
RM
such that
 
 |xf|Rx,Mx 
we say that f(x) =1
(2)
 
:xf
x
lim

If for every
,0
there exist M

R
 
,xfRx,Mx  
we say that
 


xf
x
lim
Infinite limits:
(1) If

 0x
,
 
1a;a
x
1

(2) If

 0x
,
 
1a;0a
x
1

(3) If

 0x
,
 
1a0;0a
x
1

(4) If

 0x
,
 
1a0;a
x
1

Theorem :
 


xf
ax
lim
and only if for every sequence
    
aa
x
lim
,fda,a
nnn



Impliess
 


n
af
n
lim
Important limits :
(1)
1|r|;0r
x
lim
n


(2)
 
ex1
0x
lim
x
1


(Where e = is an irrational number and
3e2 
)
(3)
 
1Ra;alog
h
1a
0h
lim
e
h




(4)
1
h
1e
0h
lim
h



Page 7
322
Continuity : Let f be a function defined on an interval (a, b) containing c.
 
b,ac
contain-
ing
 
xf
cx
lim

exists and is equal to f(c). then we say f(c), then we say f is continous at x = c.
If f is defined at isolated points, we say it is continous at that point consequently a function
defined on a finite set {x
1
,x
2
, x
3
, .......x
n
} is continous.
Continuity of a function on [a, b]
If f is defined on [a, b] , then f is continous on [a, b] if
(1) f is continous at every point (a, b)
(2)
   
afxf
ax
lim



(3)
   
bfxf
bx
lim



N.B. :
 
 
xxf 
yLku
 
 
xxf 
are continous
zRx 
and discontinous for all
zn
Theorem : Let f and g be continous
 
b,ac,cx 
(1)
gf 
is continous x = c
(2) kf continous x = c
(3) f - g continous x = c
(4) f x g continous x = c
(5)
g
k
continous for x = c if
 
0cg 
)
(6)
g
f
continous for x = c if
 
0cg 
)
Some important results of continuity :
1. A rational function is continous on its domain i.e.
 
 
 
    
0aq.ah
xq
xp
ax
lim
xh
ax
lim




2. Sin and Cosine funtions are continous on R
3. Tageent and Secant and functions are continous
 














zk
2
1k2
R
Page 8
323
4. Co-tangent and Cosecant functions are continous









zk
k
R
Continuity of Composite functions :
Let
   
d,cb,a:f 
yLku
   
f,ed,c:g 
be two functions, so that gof is continous
at
 
b,ax
1

and g is continous at
   
1
f x c, d
, then gof is continous at
 
b,ax
1

By the rule of limit of a composite function
        














1
1
1
1
xgxf
xx
lim
gxfg
xx
lim
xofg,
xx
lim
Page 9
324
Question Bank
(1)
lim
2x 
2 3
3 2
4-8x + 5x - x
= ?
2x - 9x + 12 x - 4
(a)
1
3
(b)
1
3

(c) 3 (d) -3
(2)
lim
3x 
2
4
3 4 3
= ?
81
x x x
x
   

(a)
1
24 3
(b)
1
72 15
(c)
1
72 3
(d)
1
24 15
(3)
lim
3
x


1
= ?
cot2x - 4x cot2x

([x] = x)
(a) [-1.3] (b)
 
0.75
(c) [o. 75] (d)
[1.3]
3
(4)
3
lim
2 tan 2
= ?
0


sin x x
x
x
(a) 4 (b) -8 (c) -4 (d) 8
(5)
5 π 7 π
lim
sin x . co s -c o s cos x
4 4
= ?
π
π + 4 x
x -
4

(a)
1
3

(b)
35
4
(c)
1
4

(d)
1
35

(6)
lim
2
x


 
2x sin x (4k+1 sec (4 1) sin (4 1)
2 2 2
= ?
(2 ). (4 1)
2
 
   
    
 
   
   
 
 
  
 
 
co K x k x
sec k x cos k x
  



Page 10

Limit & Continuity (Solved MCQs and Notes)

Limit & Continuity (Solved MCQs and Notes)

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