Lesson 1.8: Delta-Epsilon with Special Types of Limits

We can formally define the special types of limits (one-sided, infinite, and at infinity) in a fashion similar to the delta-epsilon definition of a standard limit. The proofs follow the same general format as the example given in the previous lesson.

Here are the definitions:

  • lim(x->a-)f(x)=L means that for every positive epsilon there exists a positive delta such that a-delta is less than x is less than a implies |f(x)-L| is less than epsilon
  • lim(x->a+)f(x)=L means that for every positive epsilon there exists a positive delta such that a is less than x is less than a+delta implies |f(x)-L| is less than epsilon
  • lim(x->a)f(x)=infinity means that for every positive M there exists a positive delta such that 0 is less than |x-a| is less than delta implies f(x)>M
  • lim(x->a)f(x)=-infinity means that for every negative N there exists a positive delta such that 0 is less than |x-a| is less than delta implies f(x) is less than N
  • lim(x->infinity)f(x)=L means that for every positive epsilon there exists a positive S such that x>S implies |f(x)-L| is less than epsilon

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