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limit : alternate defination
1. this is my attempt to redefine limits in such a way that it remove's following problems. for only defination skip to point 7.
2. limit textbook defination : assuming f(c) is not defined . lim (f(x)) at c is deduced by taking value of x as close to c as possible but not c.
3. the problem 1: 'as close to as possible', what does that even mean ? how much close is possible ? how do you determine that ?
is x + ((x-c)/2) close enough ? the inclusion of the term 'possible' without defining what does it mean mathematically
is annoying . why would even we used used the word 'possible' ? it seems like the statement ask how much one is capable of getting
x close ?
so overall i hate 'possible' and the aim of this post is to remove that 'possible' .
4. the problem 2: since i don't know which when is closest, i don't know when i am wrong . since the only way to prove i am right
is to prove that i cannot be wrong, so i don't know if i am right.
lets say f(c) = 0/0 but lim (f(c)) = L , then according to current limit defination
nothing => [ [ b =/= L] => [ lim f(c) =/= b ] ]
5. there are variables (x), when they change from one value (x_1) to another value (x_2), they go through all the value between x_1 and x_2. they
dont skip a value. for example speed (s) of a ball when increases from s_1 to s_2 then s takes all values in real subset [s_1,s_2].
lets call such variable natural_variable .
6. let's define a variable y = ((x^2) - (a^2))/(x-a) AND domain_x = R .
but turns out y(a) = 0/0 . however it is not a problem or contradiction or paradox . but you can't say [ y is natural_variable ] which is a
problem because natural_variables are to be studied . or in other word y has to be a natural_variable or to be made into one.
so to solve this we need to make sure,
i. nothing implies f(a) = 0/0
ii. assign a value to f(a) , otherwise it would still be a non natural_variable.
7. limit, alternate defination :
assign lim (f(x)) at c in such a way that
for every x
f takes all value between f(x) and ( lim f(x) at c ) within x and c.
8. limit, alternate def - mathematical version :
[ lim f(x) at c = L ] <=> [ for all x in domain_x, domain_y_x n R = domain_y_x, where domain_y_x is set of all values of f(x) between x and c ]
NOTE:
i. n is set intersection sign
ii. examples of domain_y_x
domain_y_1 for c = 2:
x_i belongs to [1,2] => (domain_y_1 for c equals 2) = {f(x_1), f(x_2), ... , f(x_n-1), L}
domain_y_1 for c = 0
x_i belongs to [0,1] => (domain_y_1 for c equals 0) = {f(x_1), f(x_2), ... , f(x_n-1), L}
9. so new_lim proves why we cannot assign lim f(x) at a = 1 if f(x) = ((x^2) - (a^2))/(x-a) . because f(x) is not equal to 1/2 between
0 and 1 -a but should have because f(0) > 1/2 > f(1-a).
so here it is. the redefination serves its purpose. i hope it helps anyone having those problems.
any comment/contradiction is welcome.
1. this is my attempt to redefine limits in such a way that it remove's following problems. for only defination skip to point 7.
2. limit textbook defination : assuming f(c) is not defined . lim (f(x)) at c is deduced by taking value of x as close to c as possible but not c.
3. the problem 1: 'as close to as possible', what does that even mean ? how much close is possible ? how do you determine that ?
is x + ((x-c)/2) close enough ? the inclusion of the term 'possible' without defining what does it mean mathematically
is annoying . why would even we used used the word 'possible' ? it seems like the statement ask how much one is capable of getting
x close ?
so overall i hate 'possible' and the aim of this post is to remove that 'possible' .
4. the problem 2: since i don't know which when is closest, i don't know when i am wrong . since the only way to prove i am right
is to prove that i cannot be wrong, so i don't know if i am right.
lets say f(c) = 0/0 but lim (f(c)) = L , then according to current limit defination
nothing => [ [ b =/= L] => [ lim f(c) =/= b ] ]
5. there are variables (x), when they change from one value (x_1) to another value (x_2), they go through all the value between x_1 and x_2. they
dont skip a value. for example speed (s) of a ball when increases from s_1 to s_2 then s takes all values in real subset [s_1,s_2].
lets call such variable natural_variable .
6. let's define a variable y = ((x^2) - (a^2))/(x-a) AND domain_x = R .
but turns out y(a) = 0/0 . however it is not a problem or contradiction or paradox . but you can't say [ y is natural_variable ] which is a
problem because natural_variables are to be studied . or in other word y has to be a natural_variable or to be made into one.
so to solve this we need to make sure,
i. nothing implies f(a) = 0/0
ii. assign a value to f(a) , otherwise it would still be a non natural_variable.
7. limit, alternate defination :
assign lim (f(x)) at c in such a way that
for every x
f takes all value between f(x) and ( lim f(x) at c ) within x and c.
8. limit, alternate def - mathematical version :
[ lim f(x) at c = L ] <=> [ for all x in domain_x, domain_y_x n R = domain_y_x, where domain_y_x is set of all values of f(x) between x and c ]
NOTE:
i. n is set intersection sign
ii. examples of domain_y_x
domain_y_1 for c = 2:
x_i belongs to [1,2] => (domain_y_1 for c equals 2) = {f(x_1), f(x_2), ... , f(x_n-1), L}
domain_y_1 for c = 0
x_i belongs to [0,1] => (domain_y_1 for c equals 0) = {f(x_1), f(x_2), ... , f(x_n-1), L}
9. so new_lim proves why we cannot assign lim f(x) at a = 1 if f(x) = ((x^2) - (a^2))/(x-a) . because f(x) is not equal to 1/2 between
0 and 1 -a but should have because f(0) > 1/2 > f(1-a).
so here it is. the redefination serves its purpose. i hope it helps anyone having those problems.
any comment/contradiction is welcome.