Question

How do you find the limit of `(xsqrt(1-4x) )/ sqrt(1-9x^3)` as x approaches `-oo`?

Answer 1

`lim_(x->-oo) f(x) = -2/3`

Explanation:

Let `f(x) = (x sqrt(1-4x))/sqrt(1-9x^3)`

Then:

`(f(x))^2 = (x^2(1-4x))/(1-9x^3)`

`=(x^2-4x^3)/(1-9x^3)`

`=(1/x-4)/(1/x^3-9)`

So `lim_(x->-oo) (f(x)^2) = (-4)/(-9) = 4/9`

Note that if `x < 0` then `f(x) < 0` consistently.

So `lim_(x->-oo) f(x) = -sqrt(lim_(x->-oo) (f(x))^2) = -sqrt(4/9) = -2/3`