**i)
Check the continuity of the function f defined by
sin 2x
if x 0
f(x)=
at x = 0 (May-2016, 17)
1
if x=0
х
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Answer:
ANSWER
We have given
f(x)=
⎩
⎪
⎨
⎪
⎧
x
2
sin
x
1
, if x
=0
0, if x=0
we know that f(0)=0
We have to determine if f is a continuous function
f(x)=x
2
sin
x
1
at x
=0
Find the limit of f(x) at x=0
R.H.L=
x→(0+h)
lim
x
2
sin
x
1
=
h→0
lim
(0+h)
2
sin
0+h
1
=
h→0
lim
h
2
sin
h
1
=0
2
×sin∞
L.H.L=
x→(0−h)
lim
x
2
sin
x
1
=
x→0
lim
(−h)
2
sin
0−h
1
L.H.L=
h→0
lim
=−h
2
sin
h
1
=−0
2
sin∞=0
R.H.S=
x→0
lim
0=0
L.H.L=R.H.L=f(0)
So, the function f(x) is continuous function
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