What is the meaning of the limit of a function?

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Answer 1 · 2015-09-14T20:11:44

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

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 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

$lim_(x→1) (x^2-1)/(x-1) = 2$