Graph limits is a recently developed powerful theory in studying large (weighted) graphs from a continuous and analytical perspective. It is particularly useful when studying subgraph homomorphism density, which is closely related to graph property testing, graph parameter estimation, and many central questions in extremal combinatorics. In this talk, we will show how the perspective of graph limits helps with graph homomorphism inequalities and how to make advances in a common theme in extremal combinatorics: when is the random construction close to optimal? We will also show some hardness results for proving general theorems in graph homomorphism density inequalities.