Not necessarily. I am a BoB customer, and in my case, a CIBIL enquiry was done. Only after that, the limit was shown.
If income is less than Rs 3 L, they straight away disregard the application.
In my experience, if you are applying for BoB Card, on a card to card basis, use a higher Annual Income (must not be very high that they will doubt), and then send your Card statement and Photo.
Your profile may be ineligible for Eterna as of now. Wait till you get cards with 2-3L limit along with strong bank credits/ITR.
Then you can get Eterna.
LTF is a different story altogether.