# How do you prove the statement lim as x approaches 2 for #(x^2 - 4x + 5) = 1# using the epsilon and delta definition?

##### 1 Answer

Please see below.

#### Explanation:

The explanation has two sections. There is a preliminary analysis to find the values used in the proof, then there is a presentation of the proof itself.

**Finding the proof**

By definition,

**if and only if**

for every

for all

We have been asked to show that

So we want to make

We want:

Look at the thing we want to make small. Rewrite this, looking for the thing we control.

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

# = (x-2)^2#

In order to make this less than

**Proving our L is correct -- Writing the proof**

Claim:

Proof:

Given

Now if

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

# = (x-2)^2#

# < delta^2# #" "# (See Note below)

# = (sqrtepsilon)^2#

# = epsilon#

We have shown that for any positive

So, by the definition of limit, we have

**Note**

Since the squaring function is increasing on positive values,