Question

Prove that f(x) = (10x3 + 25x + 63) / (4x – 3) is O(x2) directly from the definition of big O. In lecture we did a big O proof for a similar rational function. To do your proof, bound the numerator from above by a simple function on x3 and bound the denominator from below by a simple function on x. That will allow you do bound f by a simple function on x2 . If you do your proof in the style of the example from lecture (which you should!) your C will end up equal to 4 and your k will be 5.

Answer 1

Solution:

Definition big O-notation:

For two functions we say that ​​​​​​, if there exists real numbers such that

Here, We have Now,

Thus, for we get

​​​​​​.

Hence, by the definition of big O-notation, we get:

Hence the result.

Note: One can also find different values of

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