Question

How do you use the epsilon delta definition to find the limit of `(2+4x)/3` as x approaches `1`?

Answer 1

First, a slightly technical point - you can't really find a limit using the `epsilon-delta` definition. What you can do is, given a number `L`, prove that it is the limit!

Explanation:

It seems obvious that

`lim_(x to 1) (2+4x)/3 = (2+4times 1)/3 =2`

Let us prove that 2, indeed, is the limit.

Recall that the `epsilon-delta` definition of the limit states that:

A number `L` is called the limit of a function `f : RR to RR` as `x to a` if `forall epsilon>0, exists delta >0` such that
`0 < |x-a| < delta implies |f(x)-L| < epsilon`

Shorn of the Greek, this means that we can keep `f(x)` as close to `L` as we want, by keeping `x` sufficiently close to `a`.

For the current problem

`f(x)-L =(2+4x)/3-2=4/3(x-1)`

so `|f(x)-L|< epsilon iff 4/3|x-1| < epsilon`

So, for any given positive `epsilon`, choose

`0 < delta < 3/4epsilon`

Then, `|x-1| < delta implies |x-1| < 3/4epsilon implies |f(x)-L|=4/3 |x-1| < epsilon`

Hence such a `delta` exists for every positive `epsilon`, and thus 2 is the limit!