# How do you prove the statement lim as x approaches 2 for #((x^2+x-6)/(x-2))=5# using the epsilon and delta definition?

##### 1 Answer

See the explanation below.

#### Explanation:

**Preliminary Analysis**

First look at

I am looking for

# = abs(((x+3)(x-2))/(x-2)-5)#

# = abs((x+3)-5)# for all#x != 2#

# = abs(x-2)# for all#x != 2#

We want to make

If we make

So now, we are ready to write the proof.

**Proof**

Given

Now if

# = abs((x+3)-5) = abs(x-2) < delta = epsilon# .

That is:

If

By the definition of limit,