# How do you use the epsilon delta definition to find the limit of #((9-4x^2)/(3+2x))# as x approaches #-1.5#?

##### 1 Answer

See the explanation section below.

#### Explanation:

The definition of limit is not really useful for *finding* limits.

It is used to *prove* that the limit is what I said it is.

**To find this limit**

If we try to find

Both the numerator and denominator are polynomials and they share a zero. That tells us that they share a factor, so the expression can be simplified.

# = 3-2x# #" "# provided that#x != -3/2#

Since the limit doesn't care what happens when

**Proving that the limit is 6**

Claim:

Proof:

Let

Now if

# = abs((3-2x)-6)# #" "#

(Observe that

# = abs((3-2x)-6) = abs(-2x-3)#

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

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

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

# < 2delta#

# = 2(epsilon/2) = epsilon#

We have shown that, for any positive

#0 < abs(x-(-3/2)) < delta# implies#abs((9-4x^2)/(3+2x) - 6) < epsilon#

Therefore, by the definition of limit,