https://cs.stackexchange.com/q/138479/118790
Check this out and the concept said in Peter Linz. I guess it is much harder to say that whether a LANGUAGE is $LL(k)$ or not while the question is much easier to answer if the grammar is given.
See the $Example$ $7.11$ in Peter Linz, the language from the arguments given by other answers to the GATE question would make it look that the language of the grammar in Example 7.11 is not LL(k) for any $k$ but then suddenly the author comes up with a bit difficult grammar and proves that the problem with the grammar in $ Example$ $7.11$ can be resolved.
So to prove that a language is not $LL(k)$ for any $k$ I feel that much harder work is required.