MATHEMATICS
CONTENTS KEY CONCEPT .............................................................. ........................................................................... ............. Page – 2 7 EXERCISE EXERCI SE – I ...............................................................................Page – 7
EXERCISE EXERCI SE – II .............................................................................Page – 9 10 EXERCISE EXERCI SE – III ............................................................................ ............................................................................ Page – 10 13 EXERCISE – IV ................................................................ ............................................................................. ............. Page – 13
EXERCISE EXERCI SE – V ........................... ................................................P age – 14 14 VI(A) .......................................................................Page – 16 16 EXERCISE – VI(A) VI(B) .......................................................................Page – 17 17 EXERCISE – VI(B) 19 ANSWER KEY ................................................................ ............................................................................. ............. Page Pa ge – 19
GAURAV TOWER TOWE R" A-10, Road No.-1, I.P.I.A., Corporate Office : "GAURAV I.P.I.A., Kota-324005 (Raj.) INDIA.
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Email:
[email protected]
KEY CONCEPTS (LIMIT) THINGS TO REMEMBER : 1. Limit of a function f(x) is said to exist as, x!a when
Lim f (x) = Lim f (x) = finite fini te quantity..
x !a "
x !a #
2.
INDETERMINANT FORMS : 0 $ , , 0 & $, 0%, $%, $ " $ and 1$ 0 $ Note : (i) We cannot plot pl ot $ on the paper. Infinity ($) is a symbol & not a number. It does not obey the laws of elementary algebra. (ii) (iii) $ + $ = $ $ × $ = $ (iv) (a/ $) = 0 if a is finite a (v) is not defined , if a ( 0. 0 (vi) a b = 0 , if & only if a = 0 or b = 0 and a & b are finite.
3.
FUNDAMENTAL THEOREMS ON LIMIT IMITS S:
Let Lim f (x) f (x) = l & Lim g (x) = m. If l & m exists then : x !a x !a
Remember Lim ) x ( a x! a
(ii) Lim f(x). g(x) = l. m x !a
(i)
f (x) ± g (x) = l ± m Lim f (x)
(iii)
Lim
(iv)
k f(x) = k Lim f(x) ; where k is a constant. constant. Lim k f(x)
(v)
f [g(x)] = f . Lim g( x ) + = f (m) f (m) ; provided f is continuous at g (x) = m. Lim f [g(x)]
x !a
x !a
f ( x ) g(g )
*
!
m
x !a
, provided m ( 0
x !a
0 / x!a
x !a
,
6
3
For exampl exampl e Lim l n (f(x) = ln Lim f ( x ) l n l (l > 0). 45 x!a 12 x !a
4.
SQUEEZE PLAY THEOREM : Limit Limit If f(x) 7 g(x) 7 h(x) 8 x & Limit x ! a f(x) = l = x ! a h(x) then x ! a g(x) = l.
5. (a)
(b) (b)
STANDARD LIMITS :
Lim
sin x
= 1 = Lim = Lim x x !0 x !0 [Where x is measured in radians] x !0
tan " x 1
tan x
x
1/x
Lim (1 + x) = e = x !0
1 Lim 0 1# + . x !$ / x ,
x
sin " x 1
= Lim x !0
x
x
h)n = 0 note however there Lim (1 – h) h !0 n !$
and Lim (1 + h )n ! $ h !0 n !$
(c)
If
Lim f(x) = 1 and Lim x !a
x !a
9 (x) = $ , then ;
Lim
Lim :f ( x );9( x ) * e x !a 9 ( x )[ f ( x ) "1] x !a
Limit, Continuity and Differentiability Differentiability of Function
[2]
(d)
If
Lim f(x) = A > 0 & Lim x !a
x !a
Lim [f(x)] x !a
(e)
Lim
a
x !0
"1
x
x
9 (x) = B (a finite quantity) then ;
9(x) = ez where z =
Lim x !a
9 (x). ln[f(x)] = eBlnA = AB
= ln a (a > 0). In particular Lim x !0
ex
"1
x
= 1
"an * n a n "1 x"a
xn
(f)
Lim
6. (a) (b) (c)
The following strategies should be born in mind for evaluating the limits: Factorisation Rationalisation or double rationalisation Use of trigonometric transformation ; appropriate substitution and using standard limits Expansion of function like Binomial expansion, exponential & logarithmic expansion, expansion of sinx , cosx , tanx should be remembered by heart & are given below :
(d)
x !a
(i) a
x
(ii) e
x
* 1#
x ln a
#
1!
x
1!
2!
*x"
(v) cos x
* 1"
2
#
2!
x3 3
(vii) tan x = x " 1
–
x3
#
x
x3
" "
4!
2x 5 15
#
"
3
4
#
3 3 x ln a
3!
# .........a < 0
# ............ ; x = R
3!
5!
#
3
3
x5
#
3!
(vi) tan x = x #
#
2
x3
x
x
x2
(iii) ln(1+ x) = x "
(iv) sin x
2!
2
x
* 1# #
2 2 x ln a
x5 5
x4 4
# .........for " 1 > x 7 1
0 ? ? # ... ; x = . " , + / 2 2 , 7!
x7
0 ? ? # ...... ; x = . " , + / 2 2 , 6!
x
6
0 ? ? # ........ ; x = . " , + / 2 2 , "
x7 7
0 ? ? # ....... ; x = . " , + / 2 2 ,
CONTINUITY THINGS TO REMEMBER : 1. A function f(x) is said to be continuous at x = c, if Limit f(x) = f(c). Symbolically x !c f is continuous at x = c if Limit f(c - h) = Limit f(c+h) = f(c). h !0 h !0 i.e. LHL at x = c = RHL at x = c equals Value of ‘f ’ at x = c. It should be noted that continuity of a function at x = a is meaningful only if the function is defined in the immediate neighbourhood of x = a, not necessarily at x = a. 2. (i)
Reasons of discontinuity: Limit f(x) does not exist x !c i.e. Limit" f(x) ( Limit f (x) x !c
x !c #
(ii)
f(x) is not defined at x= c
(iii)
Limit f(x) ( f (c) x !c Limit, Continuity and Differentiability of Function
[3]
Geometrically, the graph of the function will exhibit a break at x= c. The graph as shown is discontinuous at x = 1 , 2 and 3. 3. Types of Discontinuities : Type - 1: ( Removable type of discontinuities) In case Limit f(x) exists but is not equal to f(c) then the function is said to have a removable discontinuity x !c
or discontinuity of the first kind. In this case we can redefine the function such that Limit f(x) = f(c) & x !c
make it continuous at x= c. Removable type of discontinuity can be further classified as : (a)
MISSING POINT DISCONTINUITY : Where Limit f(x) exists finitely but f(a) is not defined. x !a 2 sinx (1 " x )(9 " x ) e.g. f(x) = has a missing point discontinuity at x = 1 , and f(x) = has a missing point x @1 " x A discontinuity at x = 0
(b)
ISOLATED POINT DISCONTINUITY : Where Limit f(x) exists & f(a) also exists but ; Limit ( f(a). x !a x !a 2 x " 16 e.g. f(x) = , x ( 4 & f (4) = 9 has an isolated point discontinuity at x = 4. x"4 if x = I
0
6 Similarly f(x) = [x] + [ – x] = 4 5 "1
if x
BI
has an isolated point discontinuity at all x = I.
Type-2: ( Non - Removable type of discontinuities)
In case Limit f(x) does not exist then it is not possible to make the function continuous by redefining it. x !c
Such discontinuities are known as non - removable discontinuity or discontinuity of the 2nd kind. Non-removable type of discontinuity can be further classified as : (a)
Finite discontinuity e.g. f(x) = x " [x] at all integral x ; f(x) = tan ( note that f(0+) = 0 ; f(0 ) = 1 ) –
(b)
Infinite discontinuity e.g. f(x) =
1 x"4
or g(x) =
1 ( x " 4)
2
"1 1
x
at x = 0 and f(x) =
at x = 4 ; f(x) = 2tanx at x =
1 1
at x = 0
1# 2 x
? 2
and f(x) =
cosx x
at x = 0. (c)
Oscillatory discontinuity e.g. f(x) = sin 1 at x = 0. x In all these cases the value of f(a) of the function at x= a (point of discontinuity) may or may not exist but Limit does not exist. x !a
Note: From the adjacent graph note that – f is continuous at x = – 1 – f has isolated discontinuity at x = 1 – f has missing point discontinuity at x = 2 – f has non removable (finite type) discontinuity at the origin. 4.
In case of dis-continuity of the second kind the non-negative difference between the value of the RHL at x = c & LHL at x = c is called THE JUMP OF DISCONTINUITY. A function having a finite number of jumps in a given interval I is called a PIECE WISE CONTINUOUS or SECTIONALLY CONTINUOUS function in this interval.
5.
All Polynomials, Trigonometrical functions, exponential & Logarithmic functions are continuous in their domains. Limit, Continuity and Differentiability of Function
[4]
6.
If f & g are two functions that are continuous at x= c then the functions defined by : F1(x) = f(x) C g(x); F2(x) = K f(x), K any real number; F 3(x) = f(x).g(x) are also continuous at x= c. Further, if g (c) is not zero, then F4(x) =
7.
f (x) g(x)
is also continuous at x= c.
The intermediate value theorem: Suppose f(x) is continuous on an interval I , and a and b are any two points of I. Then if y 0 is a number between f(a) and f(b) , their exists a number c between a and b such that f(c) = y0. The function f, being continuous on [a,b) takes on every value between f(a) and f(b)
NOTE VERY CAREFULLY THAT : (a) If f(x) is continuous & g(x) is discontinuous at x = a then the product function 9(x) = f(x). g(x) is not necessarily be discontinuous at x = a. e.g.
6sin ?x f(x) = x & g(x) = 4 50 (b)
x(0 x*0
If f(x) and g(x) both are discontinuous at x = a then the product function 9(x) = f(x). g(x) is not necessarily be discontinuous at x = a. e.g. f(x) = " g(x) =
xD0
6 1 4 "1 5
x>0
(c)
A Continuous function whose domain is closed must have a range also in closed interval.
(d)
If f is continuous at x = c & g is continuous at x = f(c) then the composite g[f(x)] is continuous at x = c. eg. f(x) =
x sin x x
2
#2
& g(x) = ExE are continuous at x = 0 , hence the composite (gof) (x) =
x sin x x2
#2
will also
be continuous at x = 0 . 7.
CONTINUITY IN AN INTERVAL :
(a)
A function f is said to be continuous in (a , b) if f is continuous at each & every point =(a , b).
(b)
A function f is said to be continuous in a closed interval :a ,b ; if :
(i) (ii)
f is continuous in the open interval (a , b) & f is right continuous at a i.e. Limit# f(x) = f(a) = a finite quantity..
(iii)
f is left continuous at b i.e. Limit f(x) = f(b) = a finite quantity..
‘
‘
’
’
x!a
x!b"
Note that a function f which is continuous in :a ,b ; possesses the following properties : (i)
If f(a) & f(b) possess opposite signs, then there exists at least one solution of the equation f(x) = 0 in t he open interval (a , b).
(ii)
If K is any real number between f(a) & f(b), then there exists at least one solution of the equation f(x) = K in the open interval (a , b).
8.
SINGLE POINT CONTINUITY: Functions which are continuous only at one point are said to exhibit single point continuity
e.g. f(x) =
x if x = Q
"x
if x B Q
and g(x) =
x if x = Q 0 if x B Q
are both continuous only at x = 0.
Limit, Continuity and Differentiability of Function
[5]
DIFFERENTIABILITY THINGS TO REMEMBER : 1. Right hand & Left hand Derivatives ;
By definition : f F(a) = Limit h !0 (i)
# h )"f ( a ) h
if it exist
The right hand derivative of f F at x = a denoted by f F(a+) is defined by : Limit
f ' (a+) = h ! 0 # (ii)
f ( a
f ( a
# h )"f ( a ) h
,
provided the limit exists & is finite. The left hand derivative : of f at x = a denoted by f F(a+) is defined by : –
Limit
f ' (a ) = h ! 0 #
f ( a
" h )" f ( a ) , "h
Provided the limit exists & is finite. We also write f F(a+) = f F+(a) & f F(a ) = f F_(a). * This geomtrically means that a unique tangent with finite slope can be drawn at x = a as shown in the figure. –
(iii)
Derivability & Continuity : (a) If f F(a) exists then f(x) is derivable at x= a ) f(x) is continuous at x = a. (b)
If a function f is derivable at x then f is continuous at x. f ( x # h )" f ( x ) For : f F(x) = Limit exists. h !0 h f ( x # h ) " f ( x ) Also f ( x # h ) " f ( x ) * .h [ h ( 0 ] h Therefore : f ( x # h ) " f ( x ) Limit [f ( x # h ) " f ( x )] = Limit .h * f '( x ).0 * 0 h !0 h !0 h
Therefore Limit h !0 [f ( x
# h )" f ( x )] = 0 ) Limit h !0
f (x+h) = f(x) ) f is continuous at x.
Note : If f(x) is derivable for every point of its domain of definition, then it is continuous in that domain. The Converse of the above result is not true : IF f IS CONTINUOUS AT x , THEN f IS DERIVABLE AT x IS NOT TRUE. 1 e.g. the functions f(x) = E x E & g(x) = x sin ; x ( 0 & g(0) = 0 are continuous at x x = 0 but not derivable at x = 0. NOTE CAREFULLY : (a) Let f F+(a) = p & f F_(a) = q where p & q are finite then : (i) p = q ) f is derivable at x = a ) f is continuous at x = a. (ii) p ( q ) f is not derivable at x = a. It is very important to note that f may be still continuous at x = a. In short, for a function f : Differentiability ) Continuity ; Continuity ) G derivability ; “
”
Non derivability ) G discontinuous (b)
;
But discontinuity ) Non derivability
If a function f is not differentiable but is continuous at x = a it geometrically implies a sharp corner at x = a. Limit, Continuity and Differentiability of Function
[6]
3.
(i) (ii)
DERIVABILITY OVER AN INTERVAL : f (x) is said to be derivable over an interval if it is derivable at each & every point of the interval f(x) is said to be derivable over the closed interval [a, b] if : for the points a and b, f F(a+) & f F(b ") exist & for any point c such that a < c < b, f F(c+) & f F(c ") exist & are equal.
NOTE : 1. If f(x) & g(x) are derivable at x = a then the functions f(x) + g(x), f(x) " g(x) , f(x).g(x) will also be derivable at x = a & if g (a) ( 0 then the function f(x)/g(x) will also be derivable at x = a. 2. If f(x) is differentiable at x = a & g(x) is not differentiable at x = a , then the product function F(x) = f(x). g(x) can still be differentiable at x = a e.g. f(x) = x & g(x) = ExEH 3. If f(x) & g(x) both are not differentiable at x = a then the product function ; F(x) = f(x). g(x) can still be differentiable at x = a e.g. f(x) = ExE & g(x) = ExE. 4. If f(x) & g(x) both are non-deri. at x = a then the sum function F(x) = f(x) + g(x) may be a differentiable function. e.g. f(x) = ExE & g(x) = "ExEH If f(x) is derivable at x = a ) G f F(x) is continuous at x = a.
6x 2 sin x1
if x
50
if
e.g. f(x) = 4 6.
(0 x*0
A surprising result : Suppose that the function f (x) and g (x) defined in the interval (x1, x2) containing the point x0, and if f is differentiable at x = x0 with f (x0) = 0 together with g is continuous as x = x 0 then the function F (x) = f (x) · g (x) is differentiable at x = x0 e.g. F (x) = sinx · x2/3 is differentiable at x = 0.
–
6100 k 3 4 I x 1 "100 5K*1 2 x "1
Q.1
Lim x !1
Q.2
Find the sum of an infinite geometric series whose first term is the limit of the function f(x) =
as x ! ? /4 and whose common ratio is the limit of the function g(x) =
Q.3
Q.5
0 p Lim . x !1 / 1 " x p Lim x !0
"
+ p, q = N q 1 " x ,
sin 4 (3 x ) 1 " cos x
q
x!
1
2
/
x"
1 # 3 tan x
Q.7 Lim J? 1 " 2 cos 2 x x!
1"
2 sin x
1" x 1 2 as x ! 1. (cos " x )
Lim (x " l n cosh x) where cosh t = x !$
et
# e" t 2
.
.
cos "1 0 . 2x 1 " x 2 -+
Q.6 (a) Lim
Q.4
1 " tan x
1
, ;
(b) Lim x!? 4
1"
sin 2x
? " 4x
2
Q.8
Lim x !0
# 15[ x ] # 56 ; (c) Lim x ! "7 sin( x # 7) sin( x # 8) where [ ] denotes the greatest integer function [ x]
2
2 2 2 2 6 x x x x 3 " cos # cos cos 1 1 " cos 8 4 2 4 2 4 2 x 5
8
4
Limit, Continuity and Differentiability of Function
[7]
0 ? # 4h - " 4 sin 0 ? # 3h - # 6 sin 0 ? # 2h - " 4 sin 0 ? # h - # sin ? + . + . + . + 3 / 3 , / 3 , / 3 , / 3 ,
sin .
Q.9
Q.10
Lim h !0
h4
Lim x x !$
2
0 .. /
x#2 x
"3
x # 3 -
++ ,
x
Q.11
# 2x 2 ) sin 1x # | x |3 #5 3 2 | x | # | x | # | x | #1
(3x
Lim x !"$
4
0 (r # 1) sin ? " r sin ? - then find { l }. (where { } denotes the fractional part function) . + I !$ * / r #1 r , n
Q.12
Q.13
If l = Lim n
r 2
Find a & b if : (i) Lim x !$
0 x 2 # 1 . " ax " b + = 0 . x #1 + / ,
Q.14 Lim [ln (1 + sin²x). cot(ln2 (1 + x))]
Q.17
If
If
0 . /
Lim
Q.15
x !0
Q.16
(ii) xLim !"$
x !1
x
2
" x # 1 " ax " b -+ = 0
(ln (1 # x ) " ln 2)(3.4 x "1 " 3x ) 1
[(7 # x ) 3
1
" (1 # 3x ) 2 ]. sin( x " 1)
2
" 33x = ln K (where k = N), find K. Lim x !0 0 x 2 sin . . 2 ++ " sin x / , ex
0 Lim . x !3 . /
2x # 3 " x -
x "1" x 2 "5 x 2 "5 x # 6
+ + x #1 " x # 1
can be expressed in the form
a b c
where a, b, c
=
N,
then find the least value of (a2 + b2 + c2).
0 1 " 1 # ax . + exists and has the value equal to l, then find the value of 3 # 1 bx x / 1 # x , 1
1
2
3
l
b
" #
Q.18
If the Lim x !0
Q.19
Let {an}, {bn}, {cn} be sequences such that (i) an + bn + cn = 2n + 1 ; (ii) anbn +bncn + cnan = 2n – 1 ; (iii) anbncn = – 1 ; (iv) an < bn < cn
Q.20
.
Then find the value of Lim (na n ) . n !$ 3 2 Let f (x) = ax + bx + cx + d and g (x) = x 2 + x – 2. If Lim x !1
f ( x ) g( x )
= 1 and Lim x!
"1
Q.21
a
f ( x )
" 2 g( x )
2
"1
sin (1" {x}).cos (1 " {x})
Let f(x) =
= 4, then find the value of
# d2 2 2 . a #b c
2{x} . (1 " {x})
then find Lim f(x) and Lim f(x), where {x} denotes the fractional x!0 " x!0#
part function. Q.22
If Lim
a ( 2 x 3 " x 2 ) # b ( x 3 # 5 x 2 " 1) " c(3x 3 # x 2 ) a (5x 4 " x ) " bx 4
x !$
# c(4x 4 # 1) # 2 x 2 # 5x
= 1, then the value of (a + b + c) can be expressed
in the lowest form as @p q A . Find the value of (p + q). Q.23
Let L =
0 1 " 4 . + K 2 n / , n *3
0 n 3 " 1 . 3 + K . + n * 2 / n # 1 ,
; M=
" (1 # n 1 ) 2 "1 , then find the value of n *1 1 # 2n $
$
$
and N =
K
L 1 + M 1 + N 1. –
–
–
Limit, Continuity and Differentiability of Function
[8]
– Q.1
f (x) is the function such that Lim x !0
f ( x )
* 1 . If
x
x (1 # a cos x ) " b sin x
Lim x !0
3
@f (x ) A
* 1 , then find the value of
a and b. Q.2
0 ?x + Lim . tan 4 , x !1 /
Q.4
Lim x !$
Q.6
0 Lim . n !$ . /
Q.8
Q.9
Q.10
Q.3
Q.5
Lim x2 sin G !n
Q.7
6 @1 # xA1 / x 3 1 Lim 4 e 12 x !0 4 5
x
n
2
# n " 1 -+ + n ,
0 cosh (? x ) ++ Lim .. x ! $ / cos (? x ) , Lim 2 x !a (a
" x 2 )2
x !1
K* r 1
(c) (d) (e)
x!$
2 n 2 # n "1
x2
where cosh t =
e
t
[(1 " x )(1 " x )(1 " x ).........(1 " x )] n#r
3
1
(b)
r
cos
? I J x K
1 / x
2
(1 " x )(1 " x 2 )(1 " x 3 )......(1 " x 2n ) 2
F H
# e"t
0 a 2 # x 2 0 a? - sin 0 ?x - -+ . " 2 sin . + . ++ . ax 2 / , / 2 , , /
1
If L = Lim
(a)
Q.12
1
x " 1 # cos x - x Lim 0 . + x !0 x / ,
2
0 x # c * 4 then find c . + / x " c
n
Q.11
tan ?x
n
2
where a is an odd integer
then show that L can be equal to
n
K ( 4 r " 2)
n! r *1
the sum of the coefficients of two middle terms in the expansion of (1 + x)2n 1. the coefficient of xn in the expansion of (1 + x) 2n. sum of the squares of the combinatorial coefficients in (1 + x)n. –
Let x0 = 2 cos
? 6
and x n =
( n #1) · 2 " xn . 2 # x n"1 , n = 1, 2, 3, .........., find Lim 2
If a, b, c and d are real constants such that Lim x !0
n !$
ax
2
# sin(bx ) # sin(cx ) # sin(dx ) 2 4 6 3x # 5 x # 7 x
= 8,
then find the value of (a + b + c + d). n
Q.13
x cot x " Let f (x) = Lim . 3n 1 sin 3 n and g (x) = x – 4 f (x). Evaluate Lim @1 # g( x ) A x !0 n !$ 3 n *1
I
0 . ln ax + + = 6, then find the value of ln @x ln a A ln. .. ln x ++ / a ,
Q.14
If Lim# x !0
Q.15
0 x 0 x - x Evaluate Lim . " x. + ++ x !$ . e / x # 1 , , /
a.
Limit, Continuity and Differentiability of Function
[9]
Q.16
0 n - 0 3nx " 1 . kx + · . + where n = N, then find the sum of all the solutions of the equation 2 .I nx . + n / k *1 , / 3 # 1 ,+ 2
If f (x) = Lim
n !$
f (x) = x
2
"2
. 2x
Q.17
Let f (x) = Lim
sin
1# x
n !$
(a) Lim x f ( x ) , x !$
Q.18
2n
1 x
#x then find
2n
(b) Lim f ( x ) , x !1
(c) Lim f ( x ) , x !0
(d) Lim f ( x ) x ! "$
Using Sandwich theorem, evaluate (a)
(b)
0 1 Lim . n !$ . / n 2 Lim n !$
1 1# n
2
1
# n
+
2
#1
2 2#n
2
1
# n
2
#2
+ ......... +
# ........... #
+ + 2 n # 2n , 1
n n#n
2
–
Q.1
Find all possible values of a and b so that f (x) is continuous for all x = R if
O L f (x) = N L M
| ax # 3 | | 3x # a | b sin 2 x x
if x 7 "1 if " 1 > x 7 0
" 2b
cos 2 x " 3
if x D ?
tan6x
Q.2
if 0 > x > ?
6 @65 Atan5x 4 The function f(x) = 4 b#2 0 a tanx 45 @1# cosx A./ b ,+
if 0>x> ?2 if x* ? 2
if
? >x>? 2
Determine the values of 'a' & 'b' , if f is continuous at x = ? /2.
Q.3
6 f (x) 4 Suppose that f (x) = x 3 – 3x2 – 4x + 12 and h(x) = 4 x " 3 5 K
, x(3 , x*3
then
(a) find all zeros of f (x) (b) find the value of K that makes h continuous at x = 3 (c) using the value of K found in (b), determine whether h is an even function.
x2
x2
x2
# #............# and y (x) = Lim y n ( x ) 1 # x 2 (1 # x 2 )2 (1 # x 2 ) n "1 n !$ Discuss the continuity of yn(x) (n = N) and y(x) at x = 0 2
Q.4
Let yn(x) = x +
Q.5
Find the number of points of discontinuity of the function f(x) = [5x] + {3x} in [0, 5] where [y] and {y} denote largest integer less than or equal to y and fractional part of y respectively.
Limit, Continuity and Differentiability of Function
[10]
1 " sin ?x
Q.6
6 Let f(x) = 4 4 5
1 # cos 2?x
1
x>
,
p, 2x " 1 4 # 2x " 1 " 2
x
*
,x
<
2 1 2 . Determine the value of p, if possible, so that the function is 1 2
continuous at x=1/2. Q.7
Let f(x) =
61 # x 43 " x 5
, 07x72 , 2> x7 3
. Determine the form of g(x) = f [f(x)] & hence find the point of
discontinuity of g, if any. ln cos x
Q.8
Let f(x) =
6 45
1# x
if x < 0
"1 e sin 4 x " 1 if x > 0 ln (1 # tan 2 x ) 4
2
Is it possible to define f(0) to make the function continuous at x = 0. If yes what is the value of f(0), if not then indicate the nature of discontinuity.
Q.9
6 @?2 " sin "1 @1 " {x}2 AA· sin "1@1 " {x}A 4 3 2 @{x} " {x} A Let f(x) = 4 ? 4 45 2
for x
(0 where {x} is the fractional part of x.
for x
*0
Consider another function g(x) ; such that g(x) = f(x) for x D 0 = 2 2 f(x) for x < 0 Discuss the continuity of the functions f(x) & g(x) at x = 0. Q.10
Find the number of ordered pair(s) (a, b) for which the function
A
f(x) = sgn ( x 2 " ax # 1) ( bx 2 " 2 bx # 1) is discontinuous at exactly one point (where a, b are integer). [Note : sgn (x) denotes signum function of x.] Q.11
Let the equations x 3 + 2x 2 + px + q = 0 and x 3 + x 2 + px + r = 0 have two roots in common and the third root of each equation are represented by P and Q respectively.
O L If f (x) = N L M
e
x log1# x
P#Q
,
a,
"1 > x > 0 x*0
2 ln 0 . e x # PQ x -+ / , , b
tan x
Q.12
is continuous at x = 0, then find the value of 2(a + b).
0 > x >1
A function f : R ! R is defined as f (x) = Lim n !$
ax
2
# bx # c # e nx 1 # c · e nx
where f is continuous on R. Find the
values of a, b and c. Q.13
Let g (x) = Lim
x n f ( x ) # h ( x ) # 1
sin (?·2 ) 2
, x ( 1 and g (1) = Lim x !1
@
x
A
be a continuous function ln sec( ?·2 x ) 2 x # 3x # 3 at x = 1, find the value of 4 g (1) + 2 f (1) – h (1). Assume that f (x) and h (x) are continuous at x = 1. n !$
n
Limit, Continuity and Differentiability of Function
[11]
Q.14
Let f (x) = Lim n !$
x
2 n "1
# ax 3 # bx 2 . 2n x #1
If f(x) is continuous for all x = R, find the bisector of angle
between the lines 2x + y – 6 = 0 and 2x – 4y + 7 = 0 which contains the point (a, b). Q.15
" a tan x f (x) = for x > 0 tan x " sin x 2 2 ln (1 # x # x ) # ln (1 " x # x ) = for x < 0, if f is continuous at x = 0, find 'a' sec x " cos x 0 x now if g (x) = ln . 2 " + cot(x a) for x ( a, a(0, a>0. If g is continuous at x=a then show that g(e 1) = / a a sin x
–
–
Q.16
Let
O L L L f (x) = N L L L M
cosec x (sin x # cos x ) ;
a 2
3
# ex # e
x
1
ex
2
3
a ex
x
# be
"? 2
–
e.
>x>0
; x
*0
;
0> x>
? 2
If f (x) is continuous at x = 0, find the value of a 2 # b 2 . Q.17
Let f (x) = x3 – x2 – 3x – 1 and h (x) =
Q.18
O tan[ x ] , La # x L Let f (x) = N3, L 6 x " tan x 3 , Lb # 4 1 3 x 5 2 M
f ( x )
where h is a rational function such that g( x ) 1 (a) it is continuous every where except when x = – 1, (b) Lim h ( x ) * $ and (c) Lim h ( x ) * . x !"1 2 x !$ Find Lim @3h ( x ) # f ( x ) " 2g ( x ) A x !0 x<0 x
*0
x>0 $
If f (x) is continuous at x = 0 then find the value of
0 a . + I r * 0 / b ,
r
.
[Note: [k] denotes the largest integer less than or equal to k.] Q.19
Q.20 (a) (b)
Let f be a real valued continuous function on R and satisfying f ( – x) – f (x) = 0 8 x = R. If f ( – 5) = 5, f ( – 2) = 4, f (3) = – 2 and f (0) = 0 then find the minimum number of zero's of the equation f(x) = 0. If g : [a, b] onto [a, b] is continuous show that there is some c = [a, b] such that g (c) = c. Let f be continuous on the interval [0, 1] to R such that f (0) = f (1). Prove that there exists a point c in 0 c # 1 60, 1 3 . + such that f (c) = f 45 2 12 / 2 ,
Limit, Continuity and Differentiability of Function
[12]
– Q.1
Discuss the continuity & differentiability of the functions. f (x) = sin x + sinExE, x = R. x
Q.2
If the function f (x) defined as then find the range of n.
Q.3
x !y
6 " 2 for x 7 0 f (x) = 4 5 x n sin 1 for x < 0
1"
is continuous but not derivable at x = 0
x
tan x " tan y
Let g(y) = Lim
2
0 # ..1 " x ++ · tan x tan y y / y ,
x
and f(x) = x2. If h(x) = Min. @f ( x ), g( x ) A , find the number
of points where h(x) is non-derivable. Q.4
Let f (0) = 0 and f ' (0) = 1. For a positive integer k, show that 1 0 x 0 x - .. f ( x) # f 0 . + # ......f . + ++ = 1 # 1 # 1 # ...... # 1 x !0 x / 2 3 k / 2 , / k , ,
Lim
Q.5
" 0 . 1x # 1x + , ; x ( 0 , f(0) = 0, test the continuity & differentiability at x = 0. Let f(x) = x e /
Q.6
If f(x)=Ex " 1E. ( [x] " ["x]) , then find f F(1+) & f F(1-) where [x] denotes greatest integer function.
Q.7
6ax 2 " b 4 "1 If f(x) = 4 45 | x |
Q.8
Let
if x
>1
if x
D1
is derivable at x = 1. Find the values of a & b.
6 x # 1 if x > 0 f(x) = 4| x " 1 | if x D 0 5
and g(x) =
6x # 1 if x > 0 4( x " 1)2 if x D 0 5
If m, n and p are respectively the number of points where the functions f, g and gof are not derivable, find the value of (m + n + p). Q.9
6"1 5x "1
Let f(x) be defined in the interval [ – 2, 2] such that f(x) = 4
,
" 27x70 & , 0>x72
g(x) = f(ExE) + Ef(x)E. Test the differentiability of g(x) in (" 2, 2). Q.10
Examine for continuity & differentiability the points x = 1 & x = 2, the function f defined by
, 07x>2 6 x [x] where [x] = greatest integer less than or equal to x. 5(x " 1) [x] , 2 7 x 7 3 6 2 x " 3 [ x ] for x D 1 Q.11 Discuss the continuity & the derivability in [0 , 2] of f(x) = 4 ? x for x > 1 54sin 2
f(x) = 4
where [ ] denote greatest integer function . Q.12
Let f (x) = [3 + 4 sin x] (where [ ] denotes the greatest integer function). If sum of all the values of 'x' k ? in [?, 2?] where f (x) fails to be differentiable, is , then find the value of k. 2 Limit, Continuity and Differentiability of Function
[13]
Q.13
The function f ( x)
6ax( x " 1) # b * 44 x " 1 45 px 2 # qx # 2
when x > 1 when 1 7 x 7 3 when x < 3
Find the values of the constants a, b, p, q so that (i) f(x) is continuous for all x (ii) f ' (1) does not exist Q.14
Let a1 and a2 be two values of a for which f (x) =
(iii) f '(x) is continuous at x = 3
Ox. ln(1 # x ) # ln (1 " x) , L sec x " cos x N LM(a 2 " 3a # 1)x # x 2 ,
x = ( "1,0) x = [0, $)
is differentiable at x = 0, then find the value of (a12 + a22). Q.15
For any real number x, let [x] denote the largest integer less than or equal to x. Let f be a real valued
O" x " [ " x ] function defined on the interval [ – 3, 3] by f (x) = N x " [ x ] M
if [ x ] is even if [ x ] is odd .
If L denotes the number of points of discontinuity and M denotes the number of points of non-derivability of f (x), then find (L + M).
61" x 4 Q.16 f(x) = 4 x # 2 45 4 " x
,
( 0 7 x 7 1)
,
( 1> x > 2 )
,
(27 x 7 4)
Discuss the continuity & differentiability of y = f [f(x)] for 0 7 x 7 4. Q.17
Q.18
Let f be a function that is differentiable every where and that has the following properties: (i) f (x + h) = f (x) · f (h) (ii) f (x) > 0 for all real x. (iii) f ' (0) = – 1 Use the definition of derivative to find f ' (x) in terms of f (x).
6 Consider the function, f (x) = 4 5 (a) (c)
Q.1
Q.2
x 2 cos
?
if x ( 0
2x
0 if x * 0 Show that f ' (0) exists and find its value(b) Show that f ' @1 3A does not exist. For what values of x, f ' (x) fails to exist.
If the function f (x) =
3x
–
# ax # a # 3 is continuous at x = x2 # x " 2
2
Determine a & b so that f is continuous at x =
2. Find f ( – 2).
–
6 1 " sin 3 x 4 3 cos 2 x 4 ? a where f(x) = 4 2 4 4 b(1 " sin x ) 4 2 5 @? " 2x A
if x > if x if x
Limit, Continuity and Differentiability of Function
* <
? 2
? 2
? 2
[14]
Q.3
(e
6 Let f (x) = 4 45
2x
# 1) " ( x # 1)(ex # e" x ) , x x (e " 1)
k ,
if x
if x ( 0
*0
If f (x) is continuous at x = 0 then find the value of k. 1 " cos 4x Q.4
6 If f (x)= 4 a 45
x
if x > 0
2
if x * 0 x
16 # x
if x < 0
"4
Find the value of 'a' if possible so that the function is continuous at x = 0. (x
Q.5
6 If f (x) = 45 k
2
# 3x " 1) tan x 2 x # 2x
if x ( 0 if x
*0
is continuous at x = 0, then find the value of k. Q.6
If f and g are continuous functions with f (5) = 5 and Lim @2f ( x ) " g( x ) A = 6 then find the value of g(5). x !5
sin x # sin 5x
Q.7
f (x) =
if x ( "
6 cos x # cos 5x 45 k 8
x
if x * "
? 4 . Find k if f is continuous at x =
?
"? 4
.
4
" 4 x " 2 x # 12
if x < 0 2 x e x sin x # 4 x # k ln 4, x 7 0
Q.8
6 If f (x) = 4 5
Q.9
6" 4 sin x # cos x 4 If f(x) = 4 a sin x # b 4 cos x # 2 5
is continuous at x = 0 then find the value of k.
7 " ?2 " ?2 > x > ?2 x D ?2
for x for for
is continuous then find a and b.
6 14 @3x 2 # 1A "$ > x 7 1 4 Q.10 Show that the function f(x) = 45 " 4x 1> x > 4 44 " x 47x>$ 5 is continuous at x = 1 and discontinuous at x = 4. Q.11
Let g (x) =
6 45
3x 2
"4
ax # b
x
#1
for x > 1 for x D 1
.
If g (x) is continuous and differentiable for all numbers in its domain then find a and b. Q.12
Let f(x) = | x |3 find whether f "(x) exists 8 real x.
Q.13
Discuss the continuity & differentiability of the functions f (x) = ExE + Ex " 1E + Ex " 2E x = R. Also given an example of the function which is continuous everywhere but not derivable at exactly two points. Limit, Continuity and Differentiability of Function
[15]
Q.14
6 1 4 A function f is defined as follows : f(x) = 41# | sin x | 42#@x" ? A2 45 2
for for for
"$>x>0 07 x> ?2 ? 7 x ># $ 2
Discuss the continuity & differentiability at x = 0 & x = ? /2. Q.15
Let g (x) =
6a x # 2 , 0 > x > 2 4b x # 2 , 2 7 x > 5 . If g (x) is derivable on (0, 5), then find (2a + b). 5 – [1 Mark, CBSE 2002]
Q.1
Show that the function f (x) = 2x – | x | is continuous at x = 0.
Q.2
Discuss the continuity of f (x) =
Q.3
O x ,x(0 L Discuss the continuity of function f (x) = N | x | #2x 2 , at x = 0. [4 Marks, CBSE 2005] LM2, x*0
Q.4
Find the relationship between a and b so that the function f defined by f (x) =
Q.5
Oax # 1, if x 7 3 Nbx # 3, if x < 3 M
O2 x " 1, x > 0 N2 x # 1, x D 0 M
[1 Mark, CBSE 2002]
at x = 0
is continuous at x = 3.
[4 Marks, CBSE 2005, 2011]
Om( x 2 " 2x), x > 0 Find the value of m such that the function f (x) = N xD0 Mcos x,
is continuous at x = 0. [4 Marks, CBSE 2006]
Q.6
Q.7
If
O x 2 " 25 L , f (x) = N x " 5 LMk ,
when x ( 5 when x * 5
[4 Marks, CBSE 2007] If f (x), defined by the following, is continuous at x = 0, find the value of a, b, and c.
f (x) =
Q.8
is continuous at x = 5, find the value of k.
O sin(a # 1) x # sin x L , if x > 0 x LL c, if x * 0 . N L x # bx 2 " x L , if x < 0 32 LM bx
[4 Marks, CBSE 2008]
Show that the function f (x) is defined by
O sin x L x # cos x, x < 0 L x*0 f (x) = N 2, L 4@1 " 1 " x A , x>0 L M x
is continuous at x = 0.
[4 Marks, CBSE 2009]
Limit, Continuity and Differentiability of Function
[16]
Q.9
Find all points of discontinuity of f, where f is defined as follows:
OL| x | #3, x 7 "3 f (x) = N " 2x , " 3 > x > 3 LM6x # 2, x D 3 Q.10
[4 Marks, CBSE 2010]
x72 OL5 ; Find the values of a and b such that the function f (x) = Nax # b; 2 > x > 10 LM21; x D 10 [4 Marks, CBSE 2011]
is a continuous function
– Q.1 (a)
The integer n for which Lim
(cos x " 1)(cos x " e x ) xn
x !0
(A) 1
(B) 2
is a finite non-zero number is
(C) 3
(D) 4
OL tan " x f (x) = N 1 (| x|"1) LM 2 1
(b)
The domain of the derivative of the function (A) R – {0}
(c)
(B) R – {1}
if | x| < 1 is
(C) R – { – 1}
(D) R – { – 1, 1}
0 f (1 # x) Let f: R ! R be such that f (1) = 3 and f F(1) = 6. The Limit . + x!0 / f (1) , (B) e1/2
(A) 1
(d)
if | x| 7 1
Ox # a f (x) = N M| x " 1|
(C) e2
if x > 0
and
if x D 0
R / x
equals
(D) e3 [JEE 2002 (screening), 3+3+3]
Ox # 1 g (x) = N 2 M( x " 1 ) # b
if x > 0 if x D 0
Where a and b are non negative real numbers. Determine the composite function gof. If (gof) (x) is continuous for all real x, determine the values of a and b. Further, for these values of a and b, is gof differentiable at x = 0? Justify your answer. [JEE 2002, 5 out of 60] Q.2 (a)
If Lim x !0 1 (A) n
sin(n x )[(a " n )n x " tan x ] x2 (B) n2 + 1
* 0 (n > 0) then the value of 'a' is equal to 2 n #1 (C)
n
(D) None
(b)
[JEE 2003 (screening)] If a function f : [ – 2a , 2a] ! R is an odd function such that f (x) = f (2a – x) for x = [a, 2a] and the left hand derivative at x = a is 0 then find the left hand derivative at x = – a. [JEE 2003(Mains) 2 out of 60]
Q.3
62 3 1 0 1 (n # 1) cos " . + " n 1 . Find the value of Lim 4 n !$ 5 ? / n , 2
[ JEE ' 2004, 2 out of 60]
Q.4 (a)
The function given by y = | x | "1 is differentiable for all real numbers except the points (A) {0, 1, – 1}
(B) ± 1
(C) 1
(D) – 1 [JEE 2005 (Screening), 3]
Limit, Continuity and Differentiability of Function
[17]
(b)
If | f(x1) – f(x2) | 7 (x1 – x2)2, for all x1, x2 = R. Find the equation of tangent to the curve y = f (x) at the point (1, 2). [JEE 2005 (Mains), 2]
Q.5
If f (x) = min. (1, x2, x3), then
[JEE 2006, 5]
(B) f F@x A < 0 , 8 x > 1 (C) f(x) is not differentiable but continuous 8 x=R (D) f(x) is not differentiable for two values of x (A) f (x) is continuous 8 x = R
Q.6
Let g(x) =
( x " 1)
n
ln cos ( x " 1) m
; 0 < x < 2, m and n are integers, m ( 0, n > 0 and let p be the left hand
derivative of | x – 1 | at x = 1. If Lim g(x) = p, then x !1#
(B) n = 1, m = – 1
(A) n = 1, m = 1
Q.7
Let L * Lim
a" a
x !0
2
x
"x " 2
x
(A) a = 2
(C) n = 2, m = 2
(D) n > 2, m = n [JEE 2008, 3]
2
4 , a < 0 . If L is finite, then
4
(B) a = 1
(C) L =
1 64
[JEE' 2009,4]
(D) L =
1 32
Q.8 1
(a)
If Lim [1 # x ln (1 # b )] x = 2b sin2S, b > 0 and S = ( – ?, ?], then the value of S is x !0 2
(A)
(b)
If
C
? 4
O" x " ? , L 2 LL f ( x ) * N" cos x, L Lx " 1, LMln x,
(B)
C
?
(C)
3
7 "?
x
6
(D)
C
? 2
2
>x70 , 71
"? 2
(C) f (x) is differentiable at x = 1 (c)
?
"?
2 0>x x <1
(A) f (x) is continuous at x =
C
then
(B) f (x) is not differentiable at x = 0 (D) f (x) is differentiable at x =
Let f : R ! R be a function such that f (x + y) = f (x) + f (y), If f(x) is differentiable at x = 0, then (A) f (x) is differentiable only in a finite interval containing zero. (B*) f (x) is continuous 8 x = R. (C*) f '(x) is constant 8 x = R. (D) f (x) is differentiable except at finitely many points.
"3 2
8 x, y = R.
Limit, Continuity and Differentiability of Function
[18]
(d)
Let f : (0, 1) ! R be defined by f (x) =
b"x
1 " bx where b is a constant such that 0 < b < 1. Then
(B) f ( f 1 on (0, 1) and f '(b) =
(C) f = f
1
–
1
–
(A*) f is not invertible on (0, 1) 1
on (0, 1) and f '(b) =
f ' (0)
(D) f 1 is differentiable on (0, 1) –
f ' (0)
[JEE 2011, 3+4+4+4]
Q.1
5050
Q.2
a = 2; r =
1 4
;S=
Q.6 (a) DNE; (b) DNE; (c) 0 Q.7 Q.12
?
Q.17
29
3
–
–
8 3 1 3
– p"q
Q.3 Q.8
1 32
(i) a =1, b = "1 (ii) a = "1 , b =
Q.18
72
1/2
–
Q.20
ln 2
1
Q.14
2
16
Q.21
Q.5
3
Q.9
Q.13
Q.19
Q.4
2
Q.10 1/2
2
1
? 2
Q.15 ,
324
?
"
9 4
1n
4
Q.16
e
Q.1
a = – 5/2, b = – 3/2 1
–
Q.7
Q.2
" 12
Q.6
e
Q.13
g (x) = sin x and l = e Q.14
Q.17
(a) 2, (b) D.N.E., (c) 0, (d) 0
e
Q.8
e
e
?2
e3
Q.3 Q.9 Q.15 Q.18
1/2
–
e
?
2 2
a
16a "1
Q.4
#4 4
c = ln2 Q.11
27
Q.22 167 Q.23 8
2 2
– -1
"2
Q.11
Q.5
? 3
Q.12
"
?2 4
24
Q.16 0 2e (a) 2; (b) 1/2
– Q.2
a = 0 ; b = "1
Q.3
(a) "2, 2, 3 (b) K = 5 (c) even
Q.1
a = 0, b = 1
Q.4
yn(x) is continuous at x = 0 for all n and y(x) is dicontinuous at x = 0
Q.5 30 Q.6 P not possible.
Q.7
g(x) = 2 + x for 0 7 x 7 1, 2 " x for 1 < x 7 2, 4 " x for 2 < x 7 3 , g is discontinuous at x = 1 & x = 2
Q.8
f(0+) = – 2 ; f(0 ) = 2 hence f(0) not possible to define
Q.9
f(0+) =
Q.10
6
Q.15
a=
–
1 e
? 2
; f(0") =
?
) f is discont. at x = 0 ; g(0 +) = g(0") = g(0) = ? /2 ) g is cont. at x = 0
4 2
Q.11
9
Q.12
Q.16
e2 + e
2
–
c = 1, a, b = R
Q.13
Q.17 g (x) = 4 (x + 1) and limit = –
5 39 4
Q.14
6x – 2y – 5 = 0
Q.18
4
Q.19
5
–
Q.1 Q.4 Q.8 Q.10 Q.11
f(x) is conti. but not derivable at x = 0 Q.2 0 < n 7 1 + f is cont. but not diff. at x = 0 Q.6 f F (1 ) = 3 , f F(1") = "1 Q.7 5 Q.9 not derivable at x = 0 & x = 1 discontinuous & not derivable at x = 1, continuous but not derivable at x = 2 f is conti. at x = 1 , 3/2 & disconti. at x = 2, f is not diff. at x =1, 3/2 , 2
Q.3 2 a= 1/2 , b = 3/2
Limit, Continuity and Differentiability of Function
[19]
Q.12 Q.16 Q.17
Q.13
a
( 1, b * 0, p *
1
and q * " 1 Q.14 5 Q.15 8 3 f is conti. but not diff. at x = 1, disconti. at x = 2 & x = 3. cont.& diff.at all other points f ' (x) = – f (x)
24
0 1" ? f ' (0) = 0, (b) f ' .. ++ = and f ' 2 3 / ,
Q.18
(a)
Q.1
–
Q.2
a = 1/2, b = 4 Q.3
Q.7
1
Q.8
ln 2
0 1# - ? . + = , .3+ 2 / , –
1
n = I
(c) x =
2n # 1
8
Q.5
Q.11
a = 4 and b = – 4
Q.12
Yes
Q.13
continuous 8 x = R, but not derivable at x = 0,1 & 2
Q.14
continuous but not differentiable at x = 0 ; differentiable & continuous at x = ? /2.
Q.15
3
–
1
Q.9
1
Q.4
a = " 1, b = 3
"1 2
Q.6
4
– Q.2
Discontinuous
Q.5
f (x) cannot be continuous at x = 0 for any value of m. Q.6
Q.7
a=
Q.10
"3
Q.3
not continuous Q.4
; b = any real value; c =
2 a = 2; b = 1
1 2
Q.9
a=
3b # 2
(a) C; (b) D; (c) C; (d) a = 1; b = 0(gof)'(0) = 0
Q.3
1"
Q.8
(a) D; (b) A, B, C, D; (c) BC; (d) A
2
?
Q.4
(a) A, (b) y – 2 = 0
Q.5
2 3
k = 10 only point of discontinuity of f(x) is x = 3.
– Q.1
3
* b#
Q.2
(a) C; (b) f F(a ) = 0
A, C
Q.6
–
C
Q.7
A, C
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