# What is the meaning of the limit of a function?

Sep 14, 2015

The statement lim_(x→a)f(x) =L means: as $x$ gets closer to $a$, $f \left(x\right)$ 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 $f \left(x\right) = \frac{{x}^{2} - 1}{x - 1}$.

If we plot the graph, it looks like this:

We can't say what the value is at $x = 1$, but it does look as if $f \left(x\right)$ 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.

|(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