# 3.2 Asymptotics

## Practice with Big O

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
$$2x^{3}$$ 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
$$2x^{2}$$ 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
$$(1.1)^{x}$$ grows strictly faster than any polynomial.

4. $$f(x) = (0.1)^{x}$$
Exercise 4
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
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
This fraction grows as fast as $$x^{4}/x^{4}=1$$.

7. $$f(x) = 2^{3 \log_{2}x^{2}}$$
Exercise 7
$$2 ^{3 \log_{2}x^{2}} = 2^{\log_{2} (x^{2})^{3}} = 2^{\log_{2}x^{2 \cdot 3}} = 2^{\log_{2}x^{6}} = x^{6}$$