# How do you prove the statement lim as x approaches 2 for # (x^2 - 3x) = -2# using the epsilon and delta definition?

##### 1 Answer

See the explanation, below.

#### Explanation:

**Preliminary Analysis**

We want to make

By choosing

We now look at

# = abs((x-1)(x-2))#

# = abs(x-1)abs(x-2)#

If we knew the size of

Let's start by making sure that

If

If follows that

Now if we ALSO make sure that

then we will have

Now we are ready to write the proof:

**Proof**

Given

(Note that #delta is positive.)

For every

we have

(

# = abs((x-1)(x-2))#

# = abs(x-1)abs(x-2)#

# < (2)(epsilon/2) = epsilon#

We have shown that for any

if

By the definition of limit,