For the function `f(x) = (e^(1/x) -1)/(e^(1//x) + 1) , x=0` , which of the following is correct .

Options

A. `lim_(x to 0) f(x)` does not exist
B. `lim_(x to 0) f(x)=1`
C. `lim_( x to 0) f(x) ` exist but f(x) is not continuous at x =0
D. f(x) is continuous at x =0

Answer

A

Explanation

`LHL = lim_(h to 0) f(0 -h) `
` lim_( h to 0) (e^(-1//h)-1)/(e^(1//h) +1) = -1`
` [ therefore lim_(h to 0) 1/(e^(1//h)) = 0]`
RHL = `lim_( h to 0) f(0+h) = lim_( h to 0) (e^(1//h)-1)/(e^(1//h)+1) `
` = lim_( h to 0) (1 - 1/(e^(1//h)))/(1 +1/(e^(1//h)) =1`
LHL `ne ` RHL
So, limit does not exist at x =0