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Byju's Answer
Standard XII
Mathematics
Properties of Inequalities
Let f, be a c...
Question
Let
f
, be a continuous function in
[
0
,
1
]
, then
lim
n
→
∞
n
∑
j
=
0
1
n
f
(
j
n
)
is
A
1
2
1
/
2
∫
0
f
(
x
)
d
x
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B
1
∫
1
/
2
f
(
x
)
d
x
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C
1
∫
0
f
(
x
)
d
x
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D
1
/
2
∫
0
f
(
x
)
d
x
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Solution
The correct option is
C
1
∫
0
f
(
x
)
d
x
lim
n
→
∞
n
∑
j
=
0
1
n
f
(
j
n
)
1
n
→
d
x
,
j
n
→
x
,
upper limit
,
lim
n
→
∞
n
n
=
1
,
lower limit
=
lim
n
→
∞
0
n
=
0
∴
1
∫
0
f
(
x
)
d
x
Suggest Corrections
4
Similar questions
Q.
Let
f
be a non-negative function in
[
0
,
1
]
and twice differentiable in
(
0
,
1
)
.
If
x
∫
0
√
1
−
(
f
′
(
t
)
)
2
d
t
=
x
∫
0
f
(
t
)
d
t
,
0
≤
x
≤
1
and
f
(
0
)
=
0
,
then
lim
x
→
0
1
x
2
x
∫
0
f
(
t
)
d
t
Q.
Let
f
:
[
0
,
1
]
→
[
0
,
1
]
be a continuous function. Then
Q.
lim
n
→
0
1
+
n
1
+
n
2
+
n
4
Q.
Let
f
:
(
0
,
1
)
→
(
0
,
1
)
be a differential function such that
f
′
(
x
)
≠
0
for all
x
ϵ
(
0
,
1
)
and
f
(
1
2
)
=
√
3
2
.
Suppose for all
x
,
lim
t
→
x
⎛
⎝
∫
1
0
√
1
−
(
f
(
s
)
)
2
d
s
−
∫
x
0
√
1
−
(
f
(
s
)
)
2
d
s
f
(
t
)
−
f
(
x
)
⎞
⎠
=
f
(
x
)
.
Then the value of
f
(
1
4
)
belongs to :
Q.
Let
f
:
[
0
,
1
]
→
[
0
,
1
]
be a continuous function, then the equation
f
(
x
)
=
x
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