Show that `lim_(xto0^(-)) ((e^(1//x)-1)/(e^(1//x)+1))` does not exist.

1.	Show that `lim_(xto0^(-)) ((e^(1//x)-1)/(e^(1//x)+1))` does not exist.
Answer» Let `f(x)=((e^(1//x)-1)/(e^(1//x)+1)).` Then `underset(xto0^(+))limf(x)=underset(h+0)limf(0+h)=underset(hto0)limf(h)` `=lim_(hto0)((e^(1//h)-1)/(e^(1//h)+1))=underset(hto0)lim((1-(1)/(e^(1//h)))/(1+(1)/(e^(1)//h)))=((1-0)/(1+0))=1.` `underset(xto0^(-))limf(x)underset(hto0)limf(0-h)=underset(hto0)limf(-h)` `=underset(xto0)lim((e^(-1//h)-1)/(e^(-1//h)+1))=underset(hto0)lim(((1)/(e^(1//h))-1)/((1)/(e^(1//h))+1))=((0-1)/(0+1))=-1.` Thus, `underset(xto0^(+))limf(x)neunderset(xto0^(-))limf(x).` Hence, `underset(xto0)limf(x)` does not exist.

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