- #1
swampwiz
- 571
- 83
The definition says:
A finite limit exists if for every ε, there exists a δ such that if
a - δ < x < a + δ
then
| f( x ) - L | < ε
It seems that an equivalent statement would be:
A finite limit exists if for every δ that defines a domain region
a - δ < x < a + δ
that the function value is limited to a range region
| f( x ) - L | < ε
Thus it could also be said that for any domain region width δ (i.e., in the "neighborhood"), then if the function value is limited to some range region L ± ε, then the finite limit exists
Would this last statement be proper?
there exists an ε such that
A finite limit exists if for every ε, there exists a δ such that if
a - δ < x < a + δ
then
| f( x ) - L | < ε
It seems that an equivalent statement would be:
A finite limit exists if for every δ that defines a domain region
a - δ < x < a + δ
that the function value is limited to a range region
| f( x ) - L | < ε
Thus it could also be said that for any domain region width δ (i.e., in the "neighborhood"), then if the function value is limited to some range region L ± ε, then the finite limit exists
Would this last statement be proper?
there exists an ε such that