What is the meaning of the limit of a function?

1 Answer
Sep 14, 2015

The statement #lim_(x→a)f(x) =L# means: as #x# gets closer to #a#, #f(x)# gets closer to #L#.

Explanation:

The precise definition is:

For any real number #ε>0#, there exists another real number #δ>0# such that if #0<|x-a|<δ#, then #|f(x)-L|<ε#.

Consider the function #f(x) =(x^2-1)/(x-1)#.

If we plot the graph, it looks like this:

Graph

We can't say what the value is at #x=1#, but it does look as if #f(x)# approaches #2# as #x# approaches #1#.

Let's try to show that #lim_(x→1) (x^2-1)/(x-1) = 2#.

The question is, how do we get from #0<|x-1|<δ# to #|(x^2-1)/(x-1)-2| <ε#?

We must start with some value of #ε# and then find a find a corresponding value for #δ#.

Let's start with

#|(x^2-1)/(x-1) -2| =|((x+1)(x-1))/(x-1)-2| = |x+1-2| = |x-1|<ε#

The other condition is

#|x-1| < δ#

The definition fits exactly if #δ = ε#.

We have just shown that for any #ε#, there is a #δ# so that #|f(x)−2|<ε# when #0<|x−1|<δ#.

So we have shown that

#lim_(x→1) (x^2-1)/(x-1) = 2#