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Byju's Answer
Standard XII
Mathematics
Continuity in an Interval
Examine the c...
Question
Examine the continuity of the function:
f
(
x
)
=
log
100
+
log
(
0.01
+
x
)
3
x
, for
x
≠
0
=
100
3
for
x
=
0
; at
x
=
0
.
Open in App
Solution
For
x
≠
0
,
f
(
x
)
=
log
100
+
log
(
0.01
+
x
)
3
x
=
log
(
1
+
100
x
)
3
x
For continuity at
x
=
0
,
f
(
0
)
=
f
(
0
−
)
=
f
(
0
+
)
∴
lim
x
→
0
−
f
(
x
)
=
lim
x
→
0
−
log
(
1
+
100
x
)
3
x
=
lim
x
→
0
−
1
(
1
+
100
x
)
(
−
3
)
(
−
100
)
=
100
3
(Using L'Hospital's rule)
∴
lim
x
→
0
+
f
(
x
)
=
lim
x
→
0
+
log
(
1
+
100
x
)
3
x
=
lim
x
→
0
+
1
(
1
+
100
x
)
3
(
100
)
=
100
3
(Using L'Hospital's rule)
As
f
(
0
)
=
f
(
0
−
)
=
f
(
0
+
)
=
100
3
, the function is continuous at
x
=
0
.
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0
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Q.
Examine the continuity of the function:
f
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100
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