Question

What is the limit of `sqrt[x^2-2x+3 - sqrtx]` as x goes to infinity?

Answer 1

`lim_(xrarroo) sqrt[x^2-2x+3 - sqrtx] = oo`

Explanation:

`sqrt[x^2-2x+3 - sqrtx] = sqrt(x^2)sqrt(1-2/x+3/x^2=sqrtx/x^2)`

`= absx sqrt(1-2/x+3/x^2=sqrtx/x^2)`

As `x rarr oo`, the radicand goes to `1`, while `absx = x rarr oo`

Indeed, `y=x` is an oblique asymptote on the right for the graph of `f(x) = sqrt[x^2-2x+3 - sqrtx]`.