Discrete Math: Big O question

[DPXXPD]

I am at a complete loss at the math that is going on in these questions

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5. Find the smallest integer value of N such that f (x) = O (x^N ).

. . .\(\displaystyle \mbox{(a) }\, f(x)\, =\, x^7\, -\, 150\, x^6\, +\, 3\, x^3\, +\, 111\)

. . .\(\displaystyle \mbox{(b) }\, f(x)\, =\, \dfrac{x^3\, +\, 2\, x^2}{x^4\, +\, x^2\, +\, 1}\)

. . .\(\displaystyle \mbox{(c) }\, f(x)\, =\, \lceil {x} \rceil\)

. . .\(\displaystyle \mbox{(d) }\, f(x)\, =\, \lfloor {x} \rfloor\)

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Let's look at question B for now.
The answer to that question is this below:

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. . .\(\displaystyle \mbox{(b) }\, f(x) = \dfrac{x^3\, +\, 2\, x^2}{x^4\, +\, x^2\, +\, 1}:\, \mbox{ for }\, x\, \geq\, 1,\, \mbox{ we have:}\)

. . . . .\(\displaystyle \big| f(x) \big|\, \leq\, \dfrac{x^3\, +\, 2\, x^2}{x^4}\, \leq\, \dfrac{x^3\, +\, 2\, x^3}{x^4}\, =\, \dfrac{3x^3}{x^4}\, =\, 3\, x^{-1}\)

. . .\(\displaystyle \mbox{so }\, N\, =\, -1\, \mbox{ is the smallest integer that will do.}\)

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How did we come to this conclusion? What happened to the denominator of the function to change it into x^4?

 

[Deleted member 4993]

I am at a complete loss at the math that is going on in these questions

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5. Find the smallest integer value of N such that f (x) = O (x^N ).

. . .\(\displaystyle \mbox{(a) }\, f(x)\, =\, x^7\, -\, 150\, x^6\, +\, 3\, x^3\, +\, 111\)

. . .\(\displaystyle \mbox{(b) }\, f(x)\, =\, \dfrac{x^3\, +\, 2\, x^2}{x^4\, +\, x^2\, +\, 1}\)

. . .\(\displaystyle \mbox{(c) }\, f(x)\, =\, \lceil {x} \rceil\)

. . .\(\displaystyle \mbox{(d) }\, f(x)\, =\, \lfloor {x} \rfloor\)

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Let's look at question B for now.
The answer to that question is this below:

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. . .\(\displaystyle \mbox{(b) }\, f(x) = \dfrac{x^3\, +\, 2\, x^2}{x^4\, +\, x^2\, +\, 1}:\, \mbox{ for }\, x\, \geq\, 1,\, \mbox{ we have:}\)

. . . . .\(\displaystyle \big| f(x) \big|\, \leq\, \dfrac{x^3\, +\, 2\, x^2}{x^4}\, \leq\, \dfrac{x^3\, +\, 2\, x^3}{x^4}\, =\, \dfrac{3x^3}{x^4}\, =\, 3\, x^{-1}\)

. . .\(\displaystyle \mbox{so }\, N\, =\, -1\, \mbox{ is the smallest integer that will do.}\)

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How did we come to this conclusion? What happened to the denominator of the function to change it into x^4?

Do you see that :

x⁴ + x² + 1 ≥ x⁴