<s>[INST] Determine appropriate $c$ and $x_0$ for Big-O proofs. "Prove that $f(x)$ is $O(x^2)$:"
$$f(x) = \frac{x^4+2x-7}{2x^2-x-1}$$
Let $c=10$ (addition of coefficients of the numerator less the addition of coefficients of the denominator), and $x_0 = 1$ (the lowest coefficient among all elements in the function).
$$\frac{x^4+2x-7}{2x^2-x-1} \leq \frac{x^4+2x^4-7x^4}{2x^4-x^4-x^4} \leq 10x^2$$
Am I going on the right track? For any $x \geq 1$, the second fraction becomes troublesome, as you then get a division by zero. I'm not sure how to approach these questions if someone could assist me.
 [/INST] Note that
$$\frac{x^4 + 2x -7}{2x^2 - x - 1} \leq \frac{x^4 + 2x^4 -7}{2x^2 - x - 1} \leq \frac{3x^4}{2x^2 - x - 1} \leq \frac{3x^4}{x^2} = 3x^2.$$
The last inequality holds true when 
$$2x^2 - x - 1 \geq x^2, $$
that is, for all $x \geq \frac{1 + \sqrt{5}}{2}$.
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