What do courts try to determine before they rule on whether the government can limit first amendment free speech rights in a certain situation?

Whether one persons exercise of free speech would infringe on other citizens rights is indeed correct apex

The government must show probable prove that it is pertinent to limit the right. such times like emergencies, times of war which involves significant threat to national security, this right can be limited, to ensure that states secrets are not deciphered by the enemy country.