The correct option is A a=1,b=(−3)1/3
We have,
limx→0(1+a3)+8e1/x1+(1−b3)e1/x=2
[∞∞form]
⇒limx→0(1+a3)e−1/x+8e−1/x+(1−b3)=2
⇒0+80+(1−b3)=2⇒1−b3=4⇒b3=−3⇒b=(−3)1/3
Again,
limx→0(1+a3)+8e1/x1+(1−b3)e1/x=2
⇒limx→0(1+a3)+8e1/x1+4e1/x=2
⇒1+a3=2 [On comparison]
⇒a=1
Therefore, a=1,b=(−3)1/3
Hence, option 'A' is correct.