What is the limit as x approaches 3/2 of sqrt(((8x^3-27))/(4x^2-9)) ?

The limit as x approaches 3/2 of sqrt(((8x^3-27))/(4x^2-9)) is 3/(sqrt(2))

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limit as x approaches 3/2 of sqrt(((8x^3-27))/(4x^2-9))

x \to \frac{3}{2} lim (\frac{( 8 x ^{3} - 27 )}{4 x ^{2} - 9})

\frac{3}{2}

Decimal

2.12132 \dots

Solution steps

x \to \frac{3}{2} lim (\frac{8 x ^{3} - 27}{4 x ^{2} - 9})

x \to a lim [f (x)]^{b} = [x \to a lim f (x)]^{b}

With the exception of indeterminate form

= x \to \frac{3}{2} lim (\frac{8 x ^{3} - 27}{4 x ^{2} - 9})

Simplify

\frac{8 x ^{3} - 27}{4 x ^{2} - 9} : \frac{4 x ^{2} + 6 x + 9}{2 x + 3}

= x \to \frac{3}{2} lim (\frac{4 x ^{2} + 6 x + 9}{2 x + 3})

Plug in the value

x = \frac{3}{2}

= \frac{4 ( \frac{3}{2} ) ^{2} + 6 \cdot \frac{3}{2} + 9}{2 \cdot \frac{3}{2} + 3}

Simplify

\frac{4 ( \frac{3}{2} ) ^{2} + 6 \cdot \frac{3}{2} + 9}{2 \cdot \frac{3}{2} + 3} : \frac{3}{2}

= \frac{3}{2}

What is the limit as x approaches 3/2 of sqrt(((8x^3-27))/(4x^2-9)) ?
The limit as x approaches 3/2 of sqrt(((8x^3-27))/(4x^2-9)) is 3/(sqrt(2))