- #1

fishturtle1

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## Homework Statement

Verify the following assertions:

a) ##x^2 + \sqrt{x} = O(x^2)##

2. Homework Equations

2. Homework Equations

If the limit as x approaches ##\infty## of ##\frac {f(x)}{g(x)}## exists (and is finite), then ##f(x) = O(g(x))##.

## The Attempt at a Solution

Let ##\epsilon > 0##. We solve for ##\delta## such that ##x > \delta## implies ##|\frac{f(x)}{g(x)} - 1| < \epsilon##. Consider ##|\frac {x^2 + \sqrt{x}}{x^2} - 1| < \epsilon##. Then ##\frac {x^2 + \sqrt{x}}{x^2} - 1 < \epsilon##, so ##\frac {x^2 + \sqrt{x}}{x^2} < \epsilon + 1##. Taking the inverse of both sides and then taking the square root of both sides, we get ##\frac x{\sqrt{\sqrt{x} + x^2}} > \frac 1{\sqrt{\epsilon + 1}}##..

I'm trying to get to an expression "x > some expression in terms of ##\epsilon##" and then substitute that for ##\delta##.