Theorem: If f(x)→L as x→a and f(x)→L' as x→a then L=L'
Proof: to proof the theorem we first proof for f(x)→L as x→a, which is |f(x) -L|<∂ whenever n≥N₁ then we proceed to proving f(x)→L' as x→a which is |f(x) -L'|<∂ whenever n≥N₂ all these were simply obtained by using convergency of a function..
In order to equate L=L' we take m to be the maximum of N₁ and N₂, so that |f(m) -L|<∂ and |f(m) -L'|<∂ becomes true, in conclusion
|L-L'|≤|L -f(m)|+ |f(m) -L'|<2∂ so that
|L-L'|<2∂ but ∂ is any positive number whatsoever, Hence  this could only be true if L=L' which proves the statement.