Question

What is the limit of `1/sqrt(x^2 + 1)-x` as x goes to infinity?

Answer 1

`lim_(xrarroo) 1/sqrt(x^2 + 1)-x = -oo` and
`lim_(xrarroo) 1/(sqrt(x^2 + 1)-x) = oo`

Explanation:

I'm not sure the question is typed correctly, so I'll give solutions to both possibilities.

`lim_(xrarroo) 1/sqrt(x^2 + 1)-x = lim_(xrarroo) 1/sqrt(x^2 + 1)-lim_(xrarroo)x`

`= 0- lim_(xrarroo) x = -oo`

And

`lim_(xrarroo) 1/(sqrt(x^2 + 1)-x) = lim_(xrarroo) 1/((sqrt(x^2 + 1)-x)) ((sqrt(x^2 + 1)+x))/((sqrt(x^2 + 1)+x))`

`= lim_(xrarroo) (sqrt(x^2 + 1)+x)/(x^2+1-x^2)`

`= lim_(xrarroo) (sqrt(x^2 + 1)+x)`

`= lim_(xrarroo) sqrt(x^2 + 1)+ lim_(xrarroo)x`

`= oo+oo=oo`