lim x=3/2 √4x^2-9/√8x^3-27, evaluate the limits

Question

What's the answer?

Answer 1

Let x=3/2+h. √((4x^2-9)/√(8x^3-27))=√((4(3/2+h)^2-9)/√(8(x+h)^3-27)).

4(3/2+h)^2-9=12h+4h^2; 8(3/2+h)^3-27=8(27/8+27h/4+9h^2/2+h^3)-27=54h+36h^2+8h^3).

If h is very small and positive, h^2 and h^3 terms can be ignored:

The expression becomes √(12h/54h)=√(2/9)=√2/3=0.4714 approx., which is the limit as x approaches 3/2.