السلام عليكم:
هذه المسالة بدون تفكير = 1
و هي قاعدة ثابتة تساعجنا في حل مسائل اكثر تعقيدا
و اثباتها كالتالي :
نفترض ان
الزاوية theta ينتمي للفترة المفتوحة (0 ، النسبة التقريبية/2 )
لذلك
sin theta is less than theta
or
sin theta/theta is less than 1
و لكن
theta is less than or equal to tan theta
or
theta is less than sin theta/cos theta
اذا
sin theta/theta is less than cos theta
اذا نستنتج ان
cos theta is less than sin theta/theta which is less than 1 where theta does not equal to zero
و باستخدام النظرية sandwich theorem
limit as x approaches zero for cos theta is less than limit when theta aproaches zero for sin theta/theta is less than lim when theta approaches zero for 1
then
1 is less then limit as theta approaches zero for sin theta/theta is less than 1
then the limit as theta approaches to zero for the function sin theta/theta is equal to 1
