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

1 Answer

May 28, 2016

#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#

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