:: L_HOSPIT semantic presentation
theorem Th1: :: L_HOSPIT:1
theorem Th2: :: L_HOSPIT:2
theorem Th3: :: L_HOSPIT:3
theorem Th4: :: L_HOSPIT:4
theorem Th5: :: L_HOSPIT:5
theorem Th6: :: L_HOSPIT:6
theorem Th7: :: L_HOSPIT:7
for b
1, b
2 being
PartFunc of
REAL ,
REAL for b
3 being
Real holds
( ex b
4 being
Real st
( b
4 > 0 & b
1 is_differentiable_on ].b3,(b3 + b4).[ & b
2 is_differentiable_on ].b3,(b3 + b4).[ &
].b3,(b3 + b4).[ c= dom (b1 / b2) &
[.b3,(b3 + b4).] c= dom ((b1 `| ].b3,(b3 + b4).[) / (b2 `| ].b3,(b3 + b4).[)) & b
1 . b
3 = 0 & b
2 . b
3 = 0 & b
1 is_continuous_in b
3 & b
2 is_continuous_in b
3 &
(b1 `| ].b3,(b3 + b4).[) / (b2 `| ].b3,(b3 + b4).[) is_right_convergent_in b
3 ) implies ( b
1 / b
2 is_right_convergent_in b
3 & ex b
4 being
Real st
( b
4 > 0 &
lim_right (b1 / b2),b
3 = lim_right ((b1 `| ].b3,(b3 + b4).[) / (b2 `| ].b3,(b3 + b4).[)),b
3 ) ) )
theorem Th8: :: L_HOSPIT:8
for b
1, b
2 being
PartFunc of
REAL ,
REAL for b
3 being
Real holds
( ex b
4 being
Real st
( b
4 > 0 & b
1 is_differentiable_on ].(b3 - b4),b3.[ & b
2 is_differentiable_on ].(b3 - b4),b3.[ &
].(b3 - b4),b3.[ c= dom (b1 / b2) &
[.(b3 - b4),b3.] c= dom ((b1 `| ].(b3 - b4),b3.[) / (b2 `| ].(b3 - b4),b3.[)) & b
1 . b
3 = 0 & b
2 . b
3 = 0 & b
1 is_continuous_in b
3 & b
2 is_continuous_in b
3 &
(b1 `| ].(b3 - b4),b3.[) / (b2 `| ].(b3 - b4),b3.[) is_left_convergent_in b
3 ) implies ( b
1 / b
2 is_left_convergent_in b
3 & ex b
4 being
Real st
( b
4 > 0 &
lim_left (b1 / b2),b
3 = lim_left ((b1 `| ].(b3 - b4),b3.[) / (b2 `| ].(b3 - b4),b3.[)),b
3 ) ) )
theorem Th9: :: L_HOSPIT:9
theorem Th10: :: L_HOSPIT:10