1.

If `f(x)={{:(,(e^(x[x])-1)/(x+[x]),x ne 0),(,1,x=0):}` thenA. `underset(x to 0^(+))lim f(x)=-1`B. `underset(x to 0^(-))lim f(x)=(1)/(e)-1`C. f(x) is continuous at x=0D. f(x) is discontinuous at x=0

Answer» Correct Answer - D
We have
`f(x)={{:(,(e^(x[x])-1)/(x+[x]),x ne 0),(,1,x=0):}`
`f(x)={{:(,(e^(x[x])-1)/(x+[x]),-1 le x lt 0),(,1,x=0):}`
`f(x)={{:(,1,x=0),(,(e^(x)-1)/(x),0 lt x le1):}`
Clearly, we have
`underset(x to 0^(-))lim f(x)=underset(x to 0^(-))lim (e^(x-1)-1)/(x-1)=(e^(-1)-1)/(-1)=1-(1)/(e) and underset(x to 0^(+))lim f(x)=underset(x to 0^(+))lim (e^(x)-1)/(x)=1`
So, f(x) is discontinuous at x=0


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