Question

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

Answer 1

`-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`