Question

What is the formal definition of limit?

Answer 1

Precise Definitions

Finite Limit
`lim_{x to a}f(x)=L` if
for all `epsilon>0`, there exists `delta>0` such that
`0<|x-a|< delta Rightarrow |f(x)-L| < epsilon`

Infinite Limits
`lim_{x to a}f(x)=+infty` if
for all `M>0`, there exists `delta>0` such that
`0<|x-a|< delta Rightarrow f(x)>M`

`lim_{x to a}f(x)=-infty` if
for all `N<0`, there exists `delta>0` such that
`0<|x-a|< delta Rightarrow f(x) < N`