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Consider the following functions from positive integers to real numbers:

$10$, $\sqrt{n}$, $n$, $\log_{2}n$, $\frac{100}{n}$.
The CORRECT arrangement of the above functions in increasing order of asymptotic complexity is:

1. $\log_{2}n$, $\frac{100}{n}$, $10$, $\sqrt{n}$, $n$
2. $\frac{100}{n}$, $10$, $\log_{2}n$, $\sqrt{n}$, $n$
3. $10$, $\frac{100}{n}$, $\sqrt{n}$, $\log_{2}n$, $n$
4. $\frac{100}{n}$, $\log_{2}n$, $10$, $\sqrt{n}$, $n$

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Hi, if we take n=21024 , then logn gives me 1024, which is greater than 10. So why D won't be the answer?

$10$ is constant.  $\therefore$ Growth rate is $0$.

$\sqrt{n}$  grows slower than linear but faster than $\log$. (Consider $\frac {\sqrt {n^2}}{\sqrt n} =\sqrt n$, whereas $\frac {\log \left(n^2\right)}{\log n} = 2$)

$n$  : Growth rate is linear.

$\log_{2}n$  : Growth rate is logarithmic. For asymptotic growth, the base does not matter.

$\frac{100}{n}$  : Growth rate decreases with $n$.

NOTE: Please never substitute large values of $n$ in such questions. If ever you do, at least do for $2$ such values and take ratio to get the growth rate or plot a graph. Remember ${1.01}^{n} \neq O\left(n^{100}\right)$.

by Veteran (423k points)
edited
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Sir can you please explain this concept of not using large values in these type questions? That method is easy so where does it goes wrong?
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I have already given that at end rt? You have to focus on growth rate and not the value at some large point.
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Okay Thank you sir!
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@Arjun Sir

Remember ${1.01}^{n} \neq O\left(n^{100}\right)$.

why this line given?? Here one is exponential power and other one polynomial power. How these two could be related??

• $10$, Constant
• $\sqrt{n}$, Square root
• $n$, Polynomial
• $\log_{2}n$, Logarithmic
• $\frac{100}{n}$. Constant division by polynomial (clearly less than constant for every value of n >100)

Now we know  Constant division by polynomial < Constant  < Logarithmic <Square root < Polynomial So, correct order is $\frac{100}{n}$, $10$, $\log_{2}n$, $\sqrt{n}$, $n$

PS: In the above graph, for asymptotic notations any graph can be multiplied by a constant and so the $\textbf{SLOPE}$ is the important factor. If any plot is having higher slope, it'll eventually overtake the lower slope one at some value of $x$ and hence asymptotically larger than the other.

by Veteran (62.7k points)
edited by
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The graph explains the order in a nutshell. Here I m using the term win but that I dont think is right..I have been using this for my own understanding purposes..

by Active (1.9k points)

if we take a very large value of n suppose =21024

then we can follow the sequence with result

100/21024 <10 < log1024=1024 <sqrt(21024)  < 21024

so following we see B is the correct answer

by Boss (17.9k points)

100/n grows inversely with n so for very large values of n, it will become close to zero.

10 IS CONSTANT.

100/n<10

among rest of choices it is very clear that

log n < sqrt(n)<n

thus  100/n<10< log n < sqrt(n)<n

CHOICE B

by Boss (17k points)

O(100/N), O(10), O(logN), O(N1/2), O(N)

by Active (3k points)

let take  a very large N value of 21024

Now putting this in the respective  terms we get

B is the correct answer here

NOTE: More editing coming up

by Boss (17.9k points)
+1 vote
Easily comparison by taking a log of all the value and put n equals to the large value.

We get option B is correct order
by Active (1.1k points)