Aspire Faculty ID #17406 · Topic: UGC NET Computer Science June 2024 (Paper II) · Just now
UGC NET Computer Science June 2024 (Paper II)

A least integer $n$ such that $f(x)$ is $O(x^n)$ for each of the following functions. Arrange according to the value of $n$ in increasing order.

Solution

Find dominant growth for each.


A:

$3x^3+(\log x)^4$

Dominant term = $x^3$


$n=3$


B:

$(x^5+x^2+1)/(x^3+1) \approx x^2$


$n=2$


C:

$(x^3+x^2+1)/(x^2+1) \approx x$


$n=1$


D:

$(x^2+5\log x)/(x^2+1) \approx 1$


$n=0$


Thus:


D → $O(1)$

C → $O(x)$

B → $O(x^2)$

A → $O(x^3)$


Increasing order:


D, C, B, A

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