Question

How do you find the limit of `sqrt(x^2-9)/(2x-6)` as `x->-oo`?

Answer 1

`- 1/2`

Explanation:

`lim_(x to - oo) sqrt(x^2-9)/(2x-6)`

`= lim_(x to - oo) (sqrt( x^2) sqrt(1-9/x^2))/(x(2-6/x)`

`= lim_(x to - oo) (abs x sqrt(1-9/x^2))/(x(2-6/x)`

`= lim_(x to - oo) \sgn(x) (sqrt(1-9/x^2))/(2-6/x)`

`= lim_(x to - oo) \sgn(x) * (sqrt(1- lim_(x to - oo)9/x^2))/(2- lim_(x to - oo)6/x)`

`= - 1 * 1/2 = - 1/2`