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7/29/2019 Continuity&Diff Assingment
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e
)x(f ])x[1(sin ]x[ 0]x[, + 0]x[,0 ==)x(flim 6.If , then isfi0x-(a) -1 (b) 0 (c) 1 (d) none of these
(a) -2 (b) -1 (c) 0 (d) 1
7. If-2x= -)x(f ) 1e(sin-)1x(log)x(flim , then isfi
8.tan)x1(lim 1xL p2 x=
lfi-
(a) -1 (b) 0 (c) 2 p(d) p2
-1
lim-fi1x+ -p1x xcos 9. is given by
10.fi+ + (b)p2 1(c) 1 (d) 0
(a)
0xp 1
xsin1 xtan1limxeccos i
s11.(a) e (b) e-1)bxsinax(coslim +x/1 0xfi(c) 1(d) none of these
b/
a--= (a)fi0x L
1x 2fi
6 x63 xsin x
l- (c) 24-(d) 24
1 is (a) 1 (a) ab (c) eab(d) e
12. If is
limthenx25)x(f1x )1(f)x(f
24 1(b)xxsin lim +-5 1
13. Value of is
(a) 0 (b) 12 1(c)30 1(d) 120 1
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lim -+-+3 0xfix )x1log(xcosxsin1
23. If-3x 2limthen, x18- is (c) -3 1(d) none of thes
exfi L2 l i
s21. Value of is (a) -1/2 (b) 1/2 (c) 0 (d) none of these
1 )x(f=
(a) 0 (b) fi3x )3(f)x(f
9
124.sinx limx 1-|x|1 x
(a) 0 (b) 1 (c) -1 (d) none of these
2= = + then ))x(f(glim is
25. If pZn,nx,xsin )x(f and = otherwise,2)x(g= 2x,5 0x,4 2,0x,1x0x fi
(a) 5 (b) 4 (c) 2 (d) -5
limfi0x
limfi]x[cos1 ]x[cossin+-)ee(log )ax(lo
g30. Iflimfi1x23- +)1x(sin 2xx i
s26.
27.
28. is (a) 1 (b) 0 (c) does not exist (d) none of
these
ax ax is (a) 1 (b) -1 (c) 0 (d) noneof these
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(a) 2 (b) 5 (c) 3 (d) none of these =l -+2x],x[]x[ )x(f, then 'f' is continuous atx = 2 provided l is= 2x,(a) -1 (b) 0 (c) 1 (d)
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2
2
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s
)x(f= x0 + >-+ the
n39. Let 4x,8x2 4x,dy|)2y|3(
(a) f(x) is continuous as well as differentiable everywhere. (b) f(x) is
continuous everywhere but not differentiable at x = 4
(c) f(x) is neither continuous nor differentiable at x = 4. (d) 2)4(fL =40. If f(x) =cos(x2 - 2[x]) for
0 < x < 1, where
[x] denotes the
greatest integer
x , then 2 f isequal tol L p
p(d) none ofthese41. The following functions are continuous on ),0( p
x(a) tan x
(b) 01 sintdt
t (c) 9x23 x0,1p
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LEVEL - 2 (Sub jec t i ve)
1. Let to...L +f+ 2 yx= l2 )y(f)x(f for all real x and y. If )
0(f exists and equals to -1and f(0) = 1, then f(2) is equa
l1 nflim)0(fandR]1,1[:f= fi-nfi 2. If 0)0(fand nL =
l1 cos)1n( 2limfi1 n. Then thevalue of n n+ p L l
is ........
Given that 2n If= fi2bax 1x21x Limxfi --(b) If 0bax1x L+ + l= p -+0x,4])x(16[lim
0xxx2xsinx cxe)x1log(baxe = 2
.x
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limfi0x=f)y(f)x(f for all x, )1xy(,Ry and 2x L 3 1f and )1(f .. Find l1, then find lim 2xfi-2x )x(f2)2(xf(b)f(x) is differentiable functiongiven
derivative of function at x = c then 2lim)1(f)hh1(f )2(f)hh22(ffi .2
0h--+ -++
x2/a1 ,0x 2 2
1 2 1 then find the
value of 'a' and
prove that 64b =
(4 - c)
.12.= sinb )x(f-1 + L2 cxl =
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.
20 A function 'f' is
defined as f(x) = 1
1
2
| |
2
x i f x a x b x c i f
x | | | |
. If 'f'
is differentiable at++ < 12
x = 1 2
and x =
- l1 2
2
2 xx
tanxse
c.sec.
22 xx
tansec.
22 xx
tan......
sec. 22
x
, then find the values of a, b & c if possible . 21 Find a function continuous and
derivable for all x and satisfying the functional relation,
f (x + y) . f (x - y) = f(x) , where x & y are independent variables & f (0) 0 .
+ + + + 1 n n 2 3 2 n fi
where x L p
24 Evaluate: Limit 0h-+fi h x)hx(+x hx, (x > 0
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)
L E V E L - 3(Questions asked from previous Engineering Exams)xafi2 is equal to
distinct roots of ax2 +bx + c = 0, then lim a-2++-)x( )cbxax(cos1
(a) -2 2a b-a)( 2(b) b-a)( 22
(c) 2 2)( 2a b-a (d) 0
l1sinx)x(f=the
n=2.If,0x,xL)x(flimfi0x
(a) 1 (b) 0 (c) -1 (d) does not exist3.lim0xfi 2x ) x1log(xcosx= +-
4.(a)
limp fi4 x2 1
-1xcot 1xcos2= -(b) 0
(c) 1 (d) none of thes
e(a) 2 log(b)
- e a (b) 2 1
log(c) 22 1
e(d)
15.(a)
limfi0x2 1
x1x1 1a= -+2 1e a (c) a log
2 (d) none of
thes
elimx6. = -0x2fi 2x xcose
(a) 2 3(b) 2 1(c) 3 2(d) none of these
7.)x(f --+ = x px1px1+
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8.limfi2 0x= -x3sinx x5sin)x2cos1(
5-(b) = 10 3(c) 5 6(d) 6
59.(a) 3 10 32x )x(f2x, 2x= 2x,kis continuous at x = 2, then k
=(a) 16 (b) 80 (c) 32 (d) 8
10. If a, b, c, d are positive, then xfi1lim= l
L bxa 1+ ++dxc(a) ed/b(b) ec/a(c) e(c+d)/ (a+b)(d) e
11.limfi0x-xxtanxxtan ee= -
12. The value
of k which
makes =(a) 1 (b) e
(c) e - 1 (d) 0
continuous at x = 0
is
(a) 8 (b) 1
(c) -1 (d)
none of thes
e13. The value of f(0)
so that the function )
x(fxtanx2 xsinx2= is
continuous at each
point on its domain
is-1-1+ -
(a) 2 (b) 3 1(c) 3 2(d)-3 1
14. If the function 2xfor2x= 2xforx= is continuous at x = 2, thenAx)2A(x )x(f++-2
(a) A = 0 (b) A = 1 (c) A = -1 (d) none of these
(a) )x(f--+x xsin1xsin1
15. Let = . The value which should be
assigned to 'f' at x = 0 so that it is continuous
everywhere is
2 1(b) -2 (c) 2 (d) 1
16. The number of points at which the functionx|log 1)x(f= is discontinuous is
(a) 1 (b) 2 (c) 3 (d) 4
17. The value of 'b' for
which the function
1x0,4x5 )x(f2= 2x1,bx3x4 is
continuous at every
point of its domain is (a) -1 (b) 0 (c) 1 (d) 313
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= -x sin Ll 3x L 1log 42l is continuous everywhere is
18. The value of f(0) sothat )14( )x(f3 x
(a) 3(ln 4)
3
(b) 4 (ln 4)
3
(c) 12 (ln 4)
3
(d) 15 (ln 4)
3
19. Let = 3x2,x22x0,4x3 )x(f. If 'f' is continuousat x = 2, then l is
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28. If)x3(log)x3(log limfi0x= --+,kx then k is
(a) -3 1(b)3 2(c) -3 2(d) 0
29. If1lim L ++b x a2 x22 xfi,e x= l then the values of 'a' and'b' are
(a) Rb,Ra (b) Rb,1a =(c) 2b,Ra = (d) a = 1 and b =
230. If)x(f+ -xcosx xsinx= , then )x(flim2xfi is
(a) 0 (b) (c) 1 (d) none of these
(c)
lim
0x-2/
131.x limfi0x1)x1( 12= -+ (a) log 2 (b) 2 log
232.12log 2 = -fi 3x xsinxtan(d)
0(a) 1 (b) 2 (c) 2 1(d) -1
33. The function=
,xsinbx2cosa2p< px 2)x(f
iscontinuous for,x0
pthen a, barex0,xsin2axp
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36.limfi0x2 1= -x )x2cos1(
(a) 1 (b) -1 (c) 0 (d) none of these
lim37. = p0xfi 22x )xcos(sin(a) p(b)
p(c) 0x-1k)1k()1(
-1kk)1((d) p-x2 p
k k
lim -
-(d) 1 38. The left-hand derivative of f(x) = [x] sin )x(p at x = k, k an integer, is
(a) p-- )1k()1((b) p--k)1( 39. The integer 'n' for
which finx )ex)
(cos1x(cos is a finite
non-zero number i
s(a) 1 (b) 2 (c) 3 (d) 4
nxsin)xtannx)na(( lim= --2 0xfi40. If0 x, where 'n' is non-zero real number, then 'a' is equal t
o(a) 0 (b) n 1n +
(c) n (d) n +n 1
2 1l 0x 1 ,x]x2[x4 - L - L L 2 11 ,2 , 2
- L
2= axbx,
l x0
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1
43. Let )
x(f ( )
2 1- 1
sinx2sin
= l
L xand0xwhen,x2 11 ,
0 x w he n , 0 2 =
. Then f(x)
is
(a) discontinuous at x = 0 (b)
continuous in -1 , 221 L -1 2 1, 2 but differentiable in l
(c) continuous in -1 , 22 1 L ,02 1 but notdifferentiable at x = 0 (d)
differentiable only in l
44. The set of points where f(x)= |x|9 |x|+ is differentiable is
(a) )0,(- (b) ),0( (c) ),0()0,(- (d) ),(-45. At the point x = 1, the function f(x) =x3 - 1, 1 < x
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)
(
x
)
&
(
f
o
g
)
(
x
)
ar
e
b
o
t
h
c
on
t
.
f
o
r
a
l
l
x
x
+f
o
r
a
l
l
x
&
0
x
f
o
r
x
)
x(
g
=0
x
f
o
r
x
x
R (b) (gof)(x)
& (fog)(x) are
unequal
functions (c)
(gof)(x) is not
differentiable at
p>p
],0[x},xt0:)t(f{imummin)x(g,then=x,1xsin
(a)g(x)isdiscontinuousatp=x(b)g(x)iscontinuousfor ),0[x(c) g(x)isdifferentiableat p=x(d) g(x)isdifferentiablefor ),0[x+ = , ([.] denotesthe greatest
integer function),
4x1
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LEVEL - 1 (Objective)1. b 2. b 3. b 4. c 5. b 6. b 7.
c 8. b 9. b 10. b 11. d 12. a
13. d 14. b 15. d 16. d 17. c
18. c 19. d 20. d 21. a 22. c
23. dANSWERKEY
24. a 25. a 26. b 27. a 28. b 29. b 30. a 31.
d 32. c 33. b 34. c 35. b 36. a 37. a 38.
a,c,d 39. b,d 40. c 41. b,c 42. a,b,d 43. b44. b,c,d 45. a,d 46.
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a
ANSWER KEY
LEVEL -2 (Subjective)
1. f(2) = -1
2. 0
3. (a) 1a and b can have any value
(b) a = 1, b = -1
4. Not differentiable at x = 0 and 1
5.
6. a = 3, b = 12, c = 914.
15. Discontinuous at x = -1, 0, 1
16. sinx / x
17. Limit does not exist
)x(f x =18.)x(f19. a = 1, b = 0 g(f(x)) is differentiabl
e7. a = 8
8.20. a = -4, b =
0, c = 3 +ekx ==)0(fkwheree)x(f21. )
0(f L 2 b, L
3 2=loga = l=l1c,
322. tan x
23. 1/
2xx Lx2 xlnl +x 19. (a) Continuous (b) Not differentiable f= p L 3 3 1= l10. 1)1(fand24.11. (a) 2 (b) 3 12. a = 1 13. a = -1
and b = 125
A = 5, B = 2
5 and f(0) =-24
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5
ANSWER KEYLEVEL - 3 (Questions asked from previous Engineering Exams)1. c 2. b 3. a 4. b 5. a 6. a 7. b 8.
a 9. b 10. a 11. a 12. d 13. b 14. a15. d 16. c 17. a 18. c 19. c 20. a
21. b 22. d 23. b 24. c 25. a26. d
27. a 28. b 29. b 30. c 31. b 32. c
33. c 34. c 35. c 36. d 37. b 38. a39. c 40. d 41. d 42. c 43. b 44. c
45. b 46. a 47. a 48. b 49. a,b,c
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