3.2 Asymptotics

Find the least nonnegative integer, $n$, such that $f(x$) is $O(x^{n})$ when $f$ is defined by each of the expressions below.

If there is none, enter "none".

1. $f(x) = 2x^{3} + x^{2}\log x$
   Exercise 1

    Text Response  Answer:3
   $2x^{3}$ grows as fast as $x^{3}$, and $x^2\log x$ grows strictly slower than $x^{3}$.


2. $f(x) =2x^{2} + x^{3}\log x$
   Exercise 2

    Text Response  Answer:4
   $2x^{2}$ grows as fast as $x^{2}$, and $x^{3}\log x$ grows strictly faster than $x^{3}$ but strictly slower than $x^{4}$.


3. $f(x) =(1.1)^{x}$
   Exercise 3

    Text Response  Answer:none
   $(1.1)^{x}$ grows strictly faster than any polynomial.


4. $f(x) = (0.1)^{x}$
   Exercise 4

    Text Response  Answer:0
   As $x$ goes to infinity, $(0.1)^{x}$ goes to 0. So it grows strictly slower than any constant (same as a polynomial of degree 0).


5. $f(x) = \dfrac{x^{4} + x^{2} + 1}{x^{3} + 1}$
   Exercise 5

    Text Response  Answer:1
   This fraction grows as fast as $x^{4}/x^{3}=x$.


6. $f(x) = \dfrac{x^{4} + 5 \log x}{x^{4} + 1}$
   Exercise 6

    Text Response  Answer:0
   This fraction grows as fast as $x^{4}/x^{4}=1$.


7. $f(x) = 2^{3 \log_{2}x^{2}}$
   Exercise 7

    Text Response  Answer:6
   $2 ^{3 \log_{2}x^{2}} = 2^{\log_{2} (x^{2})^{3}} = 2^{\log_{2}x^{2 \cdot 3}} = 2^{\log_{2}x^{6}} = x^{6}$
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