Question

How do you find the limit of `((5x^2-2)^(1/2))/(x+3)` as x approaches `-oo`?

Answer 1

`lim_(x->-oo) (5x^2-2)^(1/2)/(x+3) = -sqrt(5)`

Explanation:

Note that if `x < 0` then `sqrt(x^2) = -x`

So:

`lim_(x->-oo) (5x^2-2)^(1/2)/(x+3) = lim_(x->-oo)(-x(5-2/x^2)^(1/2))/(x+3)`

`color(white)(lim_(x->-oo) (5x^2-2)^(1/2)/(x+3)) = lim_(x->-oo)(-(5-2/x^2)^(1/2))/(1+3/x)`

`color(white)(lim_(x->-oo) (5x^2-2)^(1/2)/(x+3)) = (-sqrt(5))/1 = -sqrt(5)`