given 'r' , a regular expression that does not include λ or ∅, What are the Conditions of 'r' so that L(r) would be infinite?
Regular expressions correspond to a simple recursive definition, which starts with the symbols from some alphabet and extends them with applications of three operators: alternation, concatenation and Kleene closure.
So ask yourself: Which of these operators can produce an infinite set?
Let $\Sigma$ be a finite set of symbols. Then the set of regular expressions $R_\Sigma$ is the smallest set of expressions which can be constructed using these rules:
Every $c \in \Sigma$ is in $R_\Sigma$.
$\lambda$ and $\emptyset$ are in $R_\Sigma$.
If $r$ and $s$ are regular expressions over the alphabet $\Sigma$ then so are:
A standard proof technique for properties of regular expressions is to apply mathematical induction to this definition.