This paper analyzes the asymptotic behaviors of Support Vector Machines (SVMs) with Gaussian (RBF) kernels, focusing on the impact of hyperparameters $C$ (penalty parameter) and $\sigma^2$ (kernel width) on the SVM classifier. The authors derive conditions under which the SVM classifier exhibits severe underfitting or overfitting, and provide insights into the behavior of the leave-one-out (loo) error. They also propose an efficient heuristic method for selecting hyperparameters that minimize generalization errors, based on the analysis of the asymptotic behaviors. The paper concludes that if complete model selection using the Gaussian kernel has been conducted, there is no need to consider linear SVMs. The analysis helps in understanding the hyperparameter space and guides the search for optimal hyperparameter values.This paper analyzes the asymptotic behaviors of Support Vector Machines (SVMs) with Gaussian (RBF) kernels, focusing on the impact of hyperparameters $C$ (penalty parameter) and $\sigma^2$ (kernel width) on the SVM classifier. The authors derive conditions under which the SVM classifier exhibits severe underfitting or overfitting, and provide insights into the behavior of the leave-one-out (loo) error. They also propose an efficient heuristic method for selecting hyperparameters that minimize generalization errors, based on the analysis of the asymptotic behaviors. The paper concludes that if complete model selection using the Gaussian kernel has been conducted, there is no need to consider linear SVMs. The analysis helps in understanding the hyperparameter space and guides the search for optimal hyperparameter values.