# Limit of a Function(Definition): Introduction to ε-δ Arguments

## Formal Definition

We say is the limit of at if, for any , there exists a such that whenever ,

## An Explanation

When most students initially confront the definition above, they are
really confused. It helps to ``unravel* the absolute values a bit.*
For example, is really the same thing as

If we then add to each term, we arrive at

In other words, we are trying to restrict to the **-neighborhood centered at** , which is the interval .
Similarly, we can rewrite to become

On its own, this would mean lies in the -neighborhood centered at , which is the interval . But what about the other requirement, that ? This means we ignore what happens at .

This neighborhood - the interval ,
minus the point - is known as a **punctured neighborhood**.

This definition can possibly be better understood through a short video.

*< Video coming soon! >*

## Proof Approach

The goal in these proofs is always the same: we need to find a ,
which will usually be expressed in terms of an arbitrary .
For example, we might have to choose a , or a ,
or even a . But in each case, we usually
build our proof __ backwards__ in what we can refer to as scratchwork.

Scratchwork begins by assuming our desired result, that

From here, we do whatever it takes (usually factoring) to change into . Once we have a statement of the form

this allows us to pick a that will work. We then just __ reverse__
the chain of equalities in our scratchwork to construct the proof.
It's easier to see in a few examples.

## Examples with Linear Functions

**Problem 1.** Using the definition of a limit, show that .

*Solution.* Looking at the statement we need to prove, we have
and . Since for any , we have , we know that for
any ,

as must be strictly positive. This means any will work. To write it out, you would proceed as follows:

**Proof.** Let be given. Choose . Then,
whenever , we have

A few quick notes about these types of proofs:

- Every one will begin with "let be given." That's because most of the time, our will be tied to the .

- We conclude the proof with a box/square, indicating we're done.

**Problem 2.** Using the definition of a limit, show that .

*Solution.* In this case, we have and .
We again begin with scratchwork, and assume our goal. If we __ knew__
that , then we would work to get an expression
that has . It goes like this:

This gives us our . We will choose a .

But that was all scratchwork, and the formal writeup looks like a bit different:

**Proof.** Let be given. Choose .
Then, whenever , we have

as required.

It should be mentioned that in cases where , and , we will get that every time.

## Examples with Quadratic Functions

**Problem 3.** Using the definition of a limit, show that .

*Solution.* Here, we have and . We again
begin with scratchwork. Suppose . We then solve
for to find

This certainly describes in terms of , but there's also an on the right hand side! This requires us to pick an ``initial </math>\delta</math>. Let's choose . Then, whenever , we have

in the manner explained in **An Explanation'. More importantly, by**
adding two to the inequality we have

Dividing by this inequality (which reverses its direction), we have

This means that for any satisfying , we know that . Thus, we can choose a , and the proof should work. There's a small problem, though - we already chose a . The way around this is to use the minimum function.

- When we write , it means to take whichever is the

least of both and .

Now, our proof can be written.

**Proof.** Let be given. Choose .
Then, whenever ,

as required.

</math>\square</math>

**Problem 4.** Using the definition of a limit, show that .

*Solution.* This time, and . We follow
the same pattern, doing the scratchwork first. Assume .
Then

Based on the previous problem, let's choose an initial . Then we have

Now adding 4 to the inequality, we have

Dividing by this inequality, we have

So we can choose Then the proof works.

**Proof.** Let be given. Choose .
Then, whenever ,

as required.

</math>\square</math>

**Problem 5.**Using the definition of a limit, show that .

*Solution.* Here, we have and .
We again begin with scratchwork, assuming . Then

Now, we again assume , so

and adding 7 to the inequality,

Again, dividing by we have

for all satisfying We can now write the proof.

**Proof.** Let be given. Choose .
Then, whenever

as required.

</math>\square</math>

## Other Examples

**Problem 6.** Using the definition of a limit, show that .

*Solution.* We have and . We start
our scratchwork as usual, assuming . Then we have

Similar to the quadratic examples, we can set an initial Then

and adding one to each term we find

This, in turn, means that

Multiplying the inequality by , we have

for all satisfying . This gives us our , and we can write the proof.

**Proof.** Let be given. Choose .
Then, whenever ,

as required. Notice that we used both of the inequalities involving and to complete the proof.

*There are many more difficult examples, but these are meant as an introduction.*