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Byju's Answer
Standard XII
Mathematics
Property 4
If fx =1-sin ...
Question
If
f
x
=
1
-
sin
x
π
-
2
x
2
,
when x ≠ π/2 and f (π/2) = λ, then f (x) will be continuous function at x = π/2, where λ =
(a) 1/8
(b) 1/4
(c) 1/2
(d) none of these
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Solution
(a)
1
8
If
f
x
is continuous at
x
=
π
2
, then
lim
x
→
π
2
f
x
=
f
π
2
lim
x
→
π
2
1
-
sin
x
π
-
2
x
2
=
f
π
2
...(1)
Suppose
π
2
-
x
=
t
, then
lim
t
→
0
1
-
sin
π
2
-
t
2
t
2
=
f
π
2
From
eq
.
(
1
)
⇒
lim
t
→
0
1
-
cos
t
4
t
2
=
f
π
2
⇒
1
4
lim
t
→
0
2
sin
2
t
2
t
2
=
f
π
2
⇒
1
4
lim
t
→
0
2
4
sin
2
t
2
t
2
4
=
f
π
2
⇒
1
8
lim
t
→
0
sin
2
t
2
t
2
4
=
f
π
2
⇒
1
8
lim
t
→
0
sin
t
2
t
2
2
=
f
π
2
⇒
f
π
2
=
λ
=
1
8
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0
Similar questions
Q.
If
f
(
x
)
=
⎧
⎪ ⎪ ⎪
⎨
⎪ ⎪ ⎪
⎩
(
1
−
sin
x
)
(
π
−
2
x
)
2
,
x
≠
π
2
λ
,
x
=
π
2
is continuous at
x
=
π
2
, then the value of
λ
is
Q.
If f(x) is continuous at
x
=
π
, where
f
(
x
)
=
√
2
+
c
o
s
x
−
1
(
π
−
x
2
)
for
x
≠
π
, find
f
(
π
)
.
Q.
Let
f
(
x
)
=
1
−
sin
x
sin
2
x
,
x
≠
π
2
. If
f
(
x
)
is continuous at
x
=
π
2
, find
f
(
π
2
)
.
Q.
If
f
(
x
)
=
1
−
sin
x
sin
2
x
,
x
≠
π
2
is continuous at
x
=
π
2
,
then the value of
f
(
π
2
)
is
Q.
If
f
(
x
)
=
1
−
sin
x
(
π
−
2
x
)
2
, for
x
≠
π
2
is continous at
x
=
π
2
, find
f
(
π
2
)
.
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