### Functions: Introduction to functions

### The notion of limit

Limit

Let #a# and #b# be real numbers and let #f# be a real function that is defined on an open interval containing #a#.

We say that #f# has **limit** #b# at #a# if #f(x)# comes closer to #b# as #x# comes closer to #a#.

In this case, we write #\textstyle\lim_{x\to a} f(x) = b# or #\displaystyle\lim_{x\to a} f(x) = b#.

To be more precise: #b# is the limit of #f# at #a# if, for every (arbitrarily small) positive number #\epsilon# a positive number #\delta# can be found with #|b-f(x)|\le\epsilon# for every #x\in\ivoo{a-\delta}{a+\delta}# with #x\ne a#.

If #b=\infty#, then #b# is the limit of #f# at #a# if there is a real number #c# such that for every (arbitrarily large) number #M# a positive number #\delta\lt |c-a|# can be found with #f(x)\ge M# for every #x\in\ivoo{a-\delta}{a+\delta}# with #x\ne a#.

If #b=-\infty#, then #b# is the limit of #f# at #a# if there is a real number #c# such that for every (arbitrarily negative) number #M# a positive number #\delta# can be found with #f(x)\le M# for every #x\in\ivoo{a-\delta}{a+\delta}# with #x\ne a#.

This follows from the fact that #f(x)=3# for every value of #x# close to (but distinct from) #8#.

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