The correct option is D (1e,1)∪(1,e)
11+lnx+11−lnx>2
The above ineuality is defined when
(i) x>0
(ii) lnx≠±1⇒x≠1e,e
Now, 21−(lnx)2>2
⇒21−(lnx)2−2>0
⇒2−2+2(lnx)21−(lnx)2>0
⇒2(lnx)21−(lnx)2>0
⇒1−(lnx)2>0[∵2(lnx)2>0 ∀ x>0 and x≠1]
⇒(lnx)2−1<0
⇒lnx∈(−1,1)
⇒x∈(1e,e)
∴x∈(1e,1)∪(1,e) (∵x≠1)