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

##### 1 Answer

See the explanation section below.

#### Explanation:

**Preliminary Analysis**

We need to make

So we begin by examining

# = 5 abs((x+1)(x-1)) = 5 abs(x+1) abs(x-1)#

We control, through our choice of

If we make sure that

(If we put a bound on the distance between

Any number will do, but is might be easiest for purpose of illustration to pick a number we haven't use yet so we can keep track of it.

Let's make sure that

If

So, adding 2 to each part, we get,

We want

#"which is" < 5(4)abs(x-1)#

# "which is"= 20 abs(x-1)#

#"we want this" < epsilon#

So let's make sure that, in addition to

we also want

Now we are ready to write the proof.

**Proof**

Claim:

Given

If

note first that

(

Furthermore, for such

# < 5(4) abs(x-1) = 20abs(x-1)#

# < 20(epsilon/20) = epsilon# .

That is: for

Therefore, by the definition on limit,

# lim_(xrarr1)5x^2 = 5#