How do you find the limit of #x-sqrt(x^2+3x)# as x approaches negative infinity?

1 Answer
Jun 26, 2016

Answer:

#-oo#

Explanation:

#lim_{x \to -oo} \ x-sqrt(x^2+3x)#

let #z = -x#

#\implies lim_{z \to oo} \ -z-sqrt(z^2-3z)#

#\implies lim_{z \to oo} \ -z + color{red}{lim_{z \to oo} -sqrt(z^2-3z)}#

the black LHS clearly goes to #-oo#

for the RHS in red, we note that #sqrt(z^2-3z) ge 0 \ forall z in [3, oo)#

therefore, the RHS clearly also goes to #-oo#

so the limit is #-oo#