Show that `("lim")_(xrarr0)` `(e^(1/x)-1)/(e^(1/x)+1) does not exist`

Options

A. -1
B. 1
C. 0
D. non-existent

Answer

D

Explanation

Let `f(x)(e^1//x-1)/(e^(1//x)+1)`.Then,
(LHL of `f(x)` at `x=0`)
` =lim_(xto0^-)f(x) =lim_(hto0) f(0-h)=lim_(hto0) (e^1//x-1)/(e^(-1//x)+1)`
`=lim_(hto0) (((1)/(e^(1//h))-1)/((1)/(e^(1//h))+1))=-1[e^(1//h)to oo rArr (1)/(e^(1//h))to0]`
and,
` [RHL of ` f(x)` at `x=0`]
`=lim_(xto0^+) f(x) =lim_(hto0) f(0+h)=lim_(hto0) (e^(1//h)-1)/(e^(1//h)+1)`
`lim_(hto0)(((1)/(e^(1//h))-1)/(1+e^(1//h)))["Dividing Nr and by "e^(1//h)]`
Clearly, `lim_(xto0^-) f(x)ne lim_(xto0^+)f(x)`.
Hence, `lim_(xto0) f(x)` does not exist.