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Question
Show that the function f given by:
`f(x)={((e^(1/x)-1)/(e^(1/x)+1),"if",x,!=,0),(-1,"if",x,=,0):}"`
is discontinuous at x = 0.
Solution
`f(x)={((e^(1/x)-1)/(e^(1/x)+1),"if",x,!=,0),(-1,"if",x,=,0):}"`
LHL: `lim_(x → 0^-) (e^(1/x) - 1)/(e^(1/x) + 1)`
= `lim_(h → 0) (e^(-1/h) - 1)/(e^(-1/h) + 1) = (0 - 1)/(0 + 1) = - 1`
RHL: `lim_(h → 0) (e^(1/h) - 1)/(e^(1/h) + 1)`
= `lim_(h → 0) (1 - e^(-1/h))/(1 + e^(-1/h)) = 1`
LHL ≠ RHL
∴ f(x) is discontinuous at x = 0.
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