Basically I need to convert these expressions into asymptotic notation. Now from what I read online, I believe number 1 should be correct, however, I'm not so sure...
1) $5x^3 - x^2 + 1$
\begin{align*}|5x^3 - x^2 + 1| &\leq 7|x^3|\\
&\leq |5x^3| + |-x^2| + 1\\
&\leq |5x^3| + |x^3| + |x^3|\\
&\leq 7|x^3| \\
&= O|x^3|\end{align*}
2) $e^{-x} + x^2$
Now for this one, I believe $x^2$ has precedence over $e^{-x}$, because $x^2$ gets much larger much faster as x-> infinity. I just wouldn't be sure what constant would work for $O(x^2)$ to make the inequality work.
3) $x|\sin(x)|$
For this one, I am not sure what to do...
4)simplify the expression $O(x^2) + O(x^3)$
Would the answer be $O(x^3)$ since $O$ represents a constant and $x^3$ increases faster?
I'm taking my classes in German and am having a hard time finding the right information. If anyone knows any good books on this subject matter, as well as stability of algorithms and that sort of stuff, please let me know.
There are multiple answer to this question depending whether you really look for $O$ notation or $\Theta$. Lets take $5x^3-x^2+1$ the correct answer in terms of $O$ notation could be $O(x^n)$ for $n \ge 3$ but also $O(e^x)$ and many others. The unique solution is $\Theta(x^3)$. Please make sure that you understand the difference between $O$ and $\Theta$ notation. Please check http://en.wikipedia.org/wiki/Big_O_notation