As robots grow more and more integrated into numerous industries, it is critical to comprehend how humans respond to their failures. This paper systematically studies how trust dynamics and system design are affected by human responses to robot failures. The three-stage survey used in the study provides a thorough understanding of human-robot interactions. While the second stage concentrates on interaction details, such as robot precision and error acknowledgment, the first stage collects demographic data and initial levels of trust. In the last phase, participants' perceptions are examined after the encounter, and trust dynamics, forgiveness, and propensity to suggest robotic technologies are evaluated. Results show that participants' trust in robotic technologies increased significantly when robots acknowledged their errors or limitations to participants and their willingness to suggest robots for activities in the future points to a favorable change in perception, emphasizing the role that direct engagement has in influencing trust dynamics. By providing useful advice for creating more sympathetic, responsive, and reliable robotic systems, the study advances the science of human-robot interaction and promotes a wider adoption of robotic technologies.