In calculus, the

**(ε, δ)-definition of limit**("epsilon-delta definition of limit") is a formalization of the notion of limit. It was given by Bernard Bolzano, in 1817, and in a less precise form by Augustin-Louis Cauchy.**Informal Statement**
Let

*ƒ*be a function. To say that
means that

*ƒ*(*x*) can be made as close as desired to*L*by making the independent variable*x*close enough, but not equal, to the value*c*.
How close is "close enough to

*c*" depends on how close one wants to make*ƒ*(*x*) to*L*. It also of course depends on which function*ƒ*is and on which number*c*is. The positive number*ε*(epsilon) is how close one wants to make*ƒ*(*x*) to*L*; one wants the distance to be less than*ε*. The positive number*δ*is how close one will make*x*to*c*; if the distance from*x*to*c*is less than*δ*(but not zero), then the distance from*ƒ*(*x*) to*L*will be less than*ε*. Thus*δ*depends on*ε*. The limit statement means that no matter how small*ε*is made,*δ*can be made small enough.
The letters

*ε*and*δ*can be understood as "error" and "distance", and in fact Cauchy used*ε*as an abbreviation for "error" in some of his work. In these terms, the error (*ε*) in the measurement of the value at the limit can be made as small as desired by reducing the distance (*δ*) to the limit point.
This definition also works for functions with more than one input value. In those cases,

*δ*can be understood as the radius of a circle or sphere or higher-dimensional analogy, in the domain of the function and centered at the point where the existence of a limit is being proven, for which every point inside produces a function value less than*ε*from the value of the function at the limit point.**Precise statement**

The (ε, δ)-definition of the limit of a function is as follows:

Let

*ƒ*be a function defined on an open interval containing*c*(except possibly at*c*) and let*L*be a real number. Then the formula
means

- for each real
*ε*> 0 there exists a real*δ*> 0 such that for all*x*with 0 < |*x*−*c*| <*δ*, we have |*ƒ*(*x*) −*L*| <*ε*,

or, symbolically,

The real inequalities exploited in the above definition were pioneered by Bolzano and Cauchy and formalized by Weierstrass.

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