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Byju's Answer
Standard XII
Mathematics
Integration to Solve Modified Sum of Binomial Coefficients
If x→ 0limα x...
Question
If
lim
x
→
0
α
x
e
x
−
β
log
e
(
1
+
x
)
+
γ
x
2
e
−
x
x
sin
2
x
=
10
,
α
,
β
,
γ
∈
R
,
then the value of
α
+
β
+
γ
is
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Solution
Using expansion, we have:
lim
x
→
0
α
x
(
1
+
x
1
!
+
x
2
2
!
+
⋯
)
−
β
(
x
−
x
2
2
+
x
3
3
+
⋯
)
+
γ
x
2
(
1
−
x
1
!
+
x
2
2
!
+
.
.
.
)
x
3
(
sin
x
x
)
2
=
10
⇒
lim
x
→
0
x
(
α
−
β
)
+
x
2
(
α
+
β
2
+
γ
)
+
x
3
(
α
2
−
β
3
−
γ
)
+
.
.
.
x
3
=
10
⇒
α
−
β
=
0
,
α
+
β
2
+
γ
=
0
,
α
2
−
β
3
−
γ
=
10
⇒
α
=
6
,
β
=
6
,
γ
=
−
9
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