Evaluate the limit of the function f(x)=sqrt[(8x^3-27)/(4x^2-9)] . x-->1,5

lim
sqrt [(8x^3-27)/(4x^2-9)]

if x -
> 1.5

We'll write the value 1.5 as a
ratio:

1.5 = 3/2

To calculate
the limit, we'll have to substitute x by the indicated value, namely
3/2.

We'll check if we'll get an indeterminacy
case.

lim sqrt [(8x^3-27)/(4x^2-9)] = sqrt (8*27/8 -
27)/(4*9/4- 9)

lim sqrt [(8x^3-27)/(4x^2-9)] = sqrt
(27-27)/(9-9)

lim sqrt [(8x^3-27)/(4x^2-9)] =
0/0, indetermination

 To calculate the limit
we'll use factorization. We notice that the numerator is a difference of
cubes:

8x^3-27 = (2x)^3 -
(3)^3

We'll apply the
formula:

a^3 - b^3 = (a-b)(a^2 + ab +
b^2)

We'll put a = 2x and b =
3

(2x)^3 - (3)^3 = (2x-3)(4x^2 + 6x +
9)

We also notice that the denominator is a difference of
squares:

4x^2-9 = (2x)^2 -
3^2

We'll apply the
formula:

a^2 - b^2 =
(a-b)(a+b)

(2x)^2 - 3^2 =
(2x-3)(2x+3)

We'll substitute the differences by their
products:

lim sqrt [(8x^3-27)/(4x^2-9)] = lim
sqrt (2x-3)(4x^2 + 6x + 9)/(2x-3)(2x+3)]

We'll
simplify:

lim sqrt [(8x^3-27)/(4x^2-9)] = lim sqrt [(4x^2 +
6x + 9)/(2x+3)]

Now, we'll substitute x by
3/2:

lim sqrt [(4x^2 + 6x + 9)/(2x+3)] = sqrt(4*9/4 + 6*3/2
+ 9)/(2*3/2 + 3)

lim sqrt [(4x^2 + 6x + 9)/(2x+3)] = sqrt
(9+9+9)/(6)

lim sqrt [(4x^2 + 6x + 9)/(2x+3)] = sqrt
27/6

lim sqrt [(4x^2 + 6x + 9)/(2x+3)] =
3sqrt18/6

lim sqrt [(4x^2 + 6x + 9)/(2x+3)] =
3sqrt2/2