What is the limit of sqrt[x^2-2x+3 - sqrtx]x22x+3x as x goes to infinity?

1 Answer
Oct 17, 2015

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].