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Byju's Answer
Standard XII
Mathematics
Derivative from First Principle
Let $x=beginc...
Question
Let
f
(
x
)
=
{
x
2
sin
1
x
,
x
≠
0
0
,
x
=
0
.
Then
A
f
′
does not exist at
x
=
0
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B
f
′
exists and is continuous at
x
=
0
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C
f
′
exists but not continuous at
x
=
0
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D
f
′
does not exist at any point
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Solution
The correct option is
C
f
′
exists but not continuous at
x
=
0
Clearly,
f
′
(
x
)
exists for all
x
≠
0
f
′
(
0
)
=
lim
h
→
0
f
(
h
)
−
f
(
0
)
h
=
lim
h
→
0
h
2
sin
1
h
h
=
lim
h
→
0
h
sin
1
h
=
0
∴
f
′
(
0
)
exists and equals
0
When
x
≠
0
,
f
′
(
x
)
=
−
cos
1
x
+
2
x
sin
1
x
lim
h
→
0
f
′
(
0
−
)
and
lim
h
→
0
f
′
(
0
+
)
do not exist.
Hence,
f
′
is not continuous at
x
=
0
Suggest Corrections
0
Similar questions
Q.
f
(
x
)
=
{
e
−
1
/
x
2
,
x
>
0
0
,
x
≤
0
, then
f
(
x
)
is
Q.
Let
f
(
x
)
=
⎧
⎨
⎩
sin
π
x
5
x
,
x
≠
0
k
,
x
=
0
. If
f
(
x
)
is continuous at
x
=
0
, the value of
k
is
Q.
Let
g
(
x
)
=
1
+
x
−
[
x
]
,
f
(
x
)
=
⎧
⎨
⎩
−
1
,
x
<
0
0
,
x
=
0
1
,
x
>
0
.
Then for all
x
,
f
(
g
(
x
)
)
is
(where
[
.
]
is greatest integer function)
Q.
If
f
(
x
)
=
{
x
+
{
x
}
+
x
sin
{
x
}
;
x
≠
0
0
;
x
=
0
where
{
x
}
denotes the fractional part function, then
Q.
Let
f
(
x
)
=
⎧
⎨
⎩
α
(
x
)
sin
π
x
2
f
o
r
x
≠
0
1
f
o
r
x
=
0
where
α
(
x
)
is such that
lim
x
→
0
|
α
(
x
)
|
=
∞
Then the function
f
(
x
)
is continuous at
x
=
0
if
α
(
x
)
is chosen as
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