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Question

Show that the function is continuous at x=0, if f(x)=sin3xtan2x,x<0

=log(1+3x)e2x1,x>0

=32,x=0

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Solution

f(x)=sin3xtan2x,x<0
=log(1+3x)e2x1,x>0
=32,x=0
Checking continuity at x=0
limx0+f(x)=limh0log(1+3(0+h))e2(0+h)1=limh0(13h+1)(3)2e2h=321 (By L' Hospital Rule)
limx0f(x)=limh0sin3(0h)tan2(0h)=limh0(cos3h)(3)(sec22h).(2)=322 (By L' Hospital Rule)
f(0)=323
From 1,2&3
limx0+f(x)=limx0f(x)=f(0)=32
f(x) is continuous at x=0

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