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

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To evaluate the function f(x) = sqrt[(8x^3-27)/(4x^2-9) ]as x --> 1.5.

If we put x= 1.5 we get f(1.5) = sqrt[(8*1.5^3-27)/(4*1.5^2-9) ] we get 0/0 form.

Therefore we divide both numerator and denominator by (x-1.5) or 2x-3 which is a factor of both numerator and denominator by the remainder theorem.

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

= (2x)^2 +2x*3+3^2 = 4x^2 +6x +9.

(4x^2-9)/(2x-3) = 2x+3,

Therefore ,

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

Now we take the limit as x --> 1.5

Lt sqrt[(8x^3-27)/(4x^2-9)] = Lt sqrt[4x^2+6x+9)/2x+3)]= sqrt {4*1.5^2+6(4.5)+9)/(2(1.5)+3)} = sqrt{27/6} = sqrt4.5 = sqrt(9/2) = sqrt18/2 = 3sqrt2/2

Therefore lt [(8x^3-27)/(4x^2-9)] = (3sqrt2)/2

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**