Question

Find the limit?

Answer 1

`-3`

Explanation:

`x+sqrt(x^2+6x)=(x+sqrt(x^2+6x))/1`

Multiply by the conjugate of `color(white)(88)x+sqrt(x^2+6x)`

`color(blue)(x-sqrt(x^2+6x))`

`((x-sqrt(x^2+6x))(x+sqrt(x^2+6x)))/(x-sqrt(x^2+6x))`

`(x^2-x^2-6x)/(x-sqrt(x^2+6x))=(-6x)/(x-sqrt(x^2+6x))`

Factor the radicand: ( Expression inside the radical )

`sqrt(x^2+6x)=sqrt(x^2(1+6/x)) =xsqrt(1+6/x)`

`(-6x)/(x+xsqrt(1+6/x)`

Divide by `color(white)(88)x`:

`(-6x/x)/(x/x+(xsqrt(1+6/x))/x`

Cancelling:

`(-6cancel(x)/cancel(x))/(cancel(x)/cancel(x)+(cancel(x)sqrt(1+6/x))/cancel(x)color(white)(88)` `=(-6)/(1+sqrt(1+6/x))`

`lim_(x->-oo)(-6)/(1+sqrt(1+6/x))=-6*lim_(x->-oo)(1)/(1+sqrt(1+6/x))`

`-6*lim_(x->-oo)(1)/(1+sqrt(1+6/x))` `=-6*(lim_(x->-oo)1)/(lim_(x->-oo)1+sqrt(1+6/x)`

`-6*lim_(x->-oo)(1)=-6`

`lim_(x->-oo)1+sqrt(1+6/x)=1+sqrt(1+0)=2`

`:.`

`(-6)/2=-3`

`lim_(x->-oo)x+sqrt(x^2+6x)=-3`