How does the concept of superposition apply in the analysis of linear electrical circuits, and what are its implications on circuit analysis methodologies? Discuss the mathematical principles behind superposition and provide examples illustrating its application in solving circuit problems. Additionally, explore the limitations of superposition and situations where it may not be applicable in practical circuit design. Finally, discuss alternative methods or techniques that can be employed in scenarios where superposition cannot be utilized effectively. Your answer should demonstrate a comprehensive understanding of the theoretical foundations of superposition and its significance in electrical engineering practice, along with critical analysis of its limitations and alternatives.