Question

How do you prove that the limit of `5 - 2x` as x approaches 2 is equal to 1 using the epsilon delta proof?

Answer 1

Given a positive `epsilon`, let `delta = epsilon/2`. Note that `delta` is positive.

Now for every `x` with `0 < abs(x-2) < delta`, we have

`abs((5-2x)-1) = abs(4-2x)`

`= abs((-2)(x-2))`

`= abs(-2)abs(x-2)`

`= 2abs(x-2)`

`< 2delta`

`= 2(epsilon/2)`

`= epsilon`.

That is: for every `x` with `0 < abs(x-2) < delta`, we have `abs((5-2x)-1) < epsilon`.

So by the definition of limit, `lim_(xrarr2)(5-2x) = 1`.