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If f(x) = xe -( 1/mod x +1/x) for x not equal to 0 and f(0)=0 then prove that f(x) is continuous for all x but not diffrentiable at x = 0

If f(x) = xe-( 1/mod x +1/x) for x not equal to 0 and f(0)=0 then prove that f(x) is continuous for all x but not diffrentiable at x = 0 

Grade:11

1 Answers

Aditya Gupta
2075 Points
2 years ago
f(x)= xe^(-2x) when x>0
And x when x
So obviously it is continuous on the real line except possibly at zero. So we check rhl and lhl separately
Lhl at zero= 0 
Rhl at zero= limit x tends to zero plus x/e^2/x= 0/infinite= 0. Since both limit are equal, and also f(0)= 0, the function is continuous at zero as well.
Hence it is continuous.
For differentiability, f'(0-)= x/x=1
f'(0+)= Lt(xe^-2/x)/x= 0 so it is not differentiable at x=0

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