The value of `lim_(x->a)log(x-a)/(log(e^x-e^a)` is

Options

A. 0
B. 1
C. a
D. does not exist

Answer

B

Explanation

`lim_(xtoa)(log(x-a))/(log(e^(x)-a^(a)))=lim_(xtoa)(1/(x-a))/(1/(e^(x)-a^(as))e^(x))`
[by L'Hospital's rule]
`=lim_(xtoa)(e^(x)-e^(a))/(e^(x)(x-a))`
`=lim(xtoa)(e^(x))/(e^(x)(x-a)+e^(x))`
[again by L'Hospital's rule]
`=(e^(a))/(e^(a))=1`