In this paper, we extend the definition of qx-asymptotic function, for extended real-valued function that define on an infinite dimensional topological normed space without lower semicontinuity or quasi-convexity condition. As the main result, by using some asymptotic conditions, we obtain sufficient optimality conditions for existence of solutions to equilibrium problems, under weaker assumptions of continuity and convexity, when the feasible set is an unbounded subset of an infinite dimensional space. Also, as a corollary, we obtain a necessary and sufficient optimality conditions for existence of solutions to equilibrium problems with unbounded feasible set. Finally, as an application, we establish a result for existence of solutions to minimization problems.