In this survey, we delve into the integration and optimization of Large Language Models (LLMs) within edge computing environments, marking a significant shift in the artificial intelligence (AI) landscape. The paper investigates the development and application of LLMs in conjunction with edge computing, highlighting the advantages of localized data processing such as reduced latency, enhanced privacy, and improved efficiency. Key challenges discussed include the deployment of LLMs on resource-limited edge devices, focusing on computational demands, energy efficiency, and model scalability. This comprehensive analysis underscores the transformative potential and future implications of combining LLMs with edge computing, paving the way for advanced AI applications across various sectors.