Article L_HOSPIT, MML version 4.99.1005
:: L_HOSPIT:th 1
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
st b1 is_continuous_in b2 &
(for b3, b4 being Element of REAL
st b3 < b2 & b2 < b4
holds ex b5, b6 being Element of REAL st
b3 < b5 & b5 < b2 & b5 in dom b1 & b6 < b4 & b2 < b6 & b6 in dom b1)
holds b1 is_convergent_in b2;
:: L_HOSPIT:th 2
theorem
for b1 being Function-like Relation of REAL,REAL
for b2, b3 being Element of REAL holds
b1 is_right_convergent_in b2 & lim_right(b1,b2) = b3
iff
(for b4 being Element of REAL
st b2 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b2 < b5 & b5 in dom b1) &
(for b4 being Function-like quasi_total Relation of NAT,REAL
st b4 is convergent & lim b4 = b2 & rng b4 c= (dom b1) /\ right_open_halfline b2
holds b1 * b4 is convergent & lim (b1 * b4) = b3);
:: L_HOSPIT:th 3
theorem
for b1 being Function-like Relation of REAL,REAL
for b2, b3 being Element of REAL holds
b1 is_left_convergent_in b2 & lim_left(b1,b2) = b3
iff
(for b4 being Element of REAL
st b4 < b2
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b2 & b5 in dom b1) &
(for b4 being Function-like quasi_total Relation of NAT,REAL
st b4 is convergent & lim b4 = b2 & rng b4 c= (dom b1) /\ halfline b2
holds b1 * b4 is convergent & lim (b1 * b4) = b3);
:: L_HOSPIT:th 4
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
st ex b3 being Neighbourhood of b2 st
b3 \ {b2} c= dom b1
for b3, b4 being Element of REAL
st b3 < b2 & b2 < b4
holds ex b5, b6 being Element of REAL st
b3 < b5 & b5 < b2 & b5 in dom b1 & b6 < b4 & b2 < b6 & b6 in dom b1;
:: L_HOSPIT:th 5
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
st ex b4 being Neighbourhood of b3 st
b1 is_differentiable_on b4 &
b2 is_differentiable_on b4 &
b4 \ {b3} c= dom (b1 / b2) &
b4 c= dom ((b1 `| b4) / (b2 `| b4)) &
b1 . b3 = 0 &
b2 . b3 = 0 &
(b1 `| b4) / (b2 `| b4) is_divergent_to+infty_in b3
holds b1 / b2 is_divergent_to+infty_in b3;
:: L_HOSPIT:th 6
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
st ex b4 being Neighbourhood of b3 st
b1 is_differentiable_on b4 &
b2 is_differentiable_on b4 &
b4 \ {b3} c= dom (b1 / b2) &
b4 c= dom ((b1 `| b4) / (b2 `| b4)) &
b1 . b3 = 0 &
b2 . b3 = 0 &
(b1 `| b4) / (b2 `| b4) is_divergent_to-infty_in b3
holds b1 / b2 is_divergent_to-infty_in b3;
:: L_HOSPIT:th 7
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
st ex b4 being Element of REAL st
0 < b4 &
b1 is_differentiable_on ].b3,b3 + b4.[ &
b2 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.[)) &
b1 . b3 = 0 &
b2 . b3 = 0 &
b1 is_continuous_in b3 &
b2 is_continuous_in b3 &
(b1 `| ].b3,b3 + b4.[) / (b2 `| ].b3,b3 + b4.[) is_right_convergent_in b3
holds b1 / b2 is_right_convergent_in b3 &
(ex b4 being Element of REAL st
0 < b4 &
lim_right(b1 / b2,b3) = lim_right((b1 `| ].b3,b3 + b4.[) / (b2 `| ].b3,b3 + b4.[),b3));
:: L_HOSPIT:th 8
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
st ex b4 being Element of REAL st
0 < b4 &
b1 is_differentiable_on ].b3 - b4,b3.[ &
b2 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.[)) &
b1 . b3 = 0 &
b2 . b3 = 0 &
b1 is_continuous_in b3 &
b2 is_continuous_in b3 &
(b1 `| ].b3 - b4,b3.[) / (b2 `| ].b3 - b4,b3.[) is_left_convergent_in b3
holds b1 / b2 is_left_convergent_in b3 &
(ex b4 being Element of REAL st
0 < b4 &
lim_left(b1 / b2,b3) = lim_left((b1 `| ].b3 - b4,b3.[) / (b2 `| ].b3 - b4,b3.[),b3));
:: L_HOSPIT:th 9
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
st ex b4 being Neighbourhood of b3 st
b1 is_differentiable_on b4 &
b2 is_differentiable_on b4 &
b4 \ {b3} c= dom (b1 / b2) &
b4 c= dom ((b1 `| b4) / (b2 `| b4)) &
b1 . b3 = 0 &
b2 . b3 = 0 &
(b1 `| b4) / (b2 `| b4) is_convergent_in b3
holds b1 / b2 is_convergent_in b3 &
(ex b4 being Neighbourhood of b3 st
lim(b1 / b2,b3) = lim((b1 `| b4) / (b2 `| b4),b3));
:: L_HOSPIT:th 10
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
st ex b4 being Neighbourhood of b3 st
b1 is_differentiable_on b4 &
b2 is_differentiable_on b4 &
b4 \ {b3} c= dom (b1 / b2) &
b4 c= dom ((b1 `| b4) / (b2 `| b4)) &
b1 . b3 = 0 &
b2 . b3 = 0 &
(b1 `| b4) / (b2 `| b4) is_continuous_in b3
holds b1 / b2 is_convergent_in b3 &
lim(b1 / b2,b3) = (diff(b1,b3)) / diff(b2,b3);