Answer to Question #280878 in Discrete Mathematics for jack

Question #280878

State TRUE or FALSE justifying your answer with proper reason.

a. 2𝑛^2 + 1 = 𝑂(𝑛^2 )

b. 𝑛^2 (1 + βˆšπ‘›) = 𝑂(𝑛^2 )

c. 𝑛^2 (1 + βˆšπ‘›) = 𝑂(𝑛^2 log 𝑛)

d. 3𝑛^2 + βˆšπ‘› = 𝑂(𝑛 + π‘›βˆšπ‘› + βˆšπ‘›)

e. βˆšπ‘› log 𝑛 = 𝑂(𝑛)


1
Expert's answer
2022-01-18T18:09:50-0500

a.

true

2𝑛2+1≀3n22𝑛^2 + 1\le 3n^2


b.

false

lim⁑nβ†’βˆžπ‘›2(1+𝑛)n2=∞\displaystyle \lim_{n\to \infin} \frac{𝑛^2 (1 + \sqrt𝑛) }{n^2}=\infin


c.

false

lim⁑nβ†’βˆžπ‘›2(1+𝑛)n2logn=∞\displaystyle \lim_{n\to \infin} \frac{𝑛^2 (1 + \sqrt𝑛) }{n^2logn}=\infin


d.

false

lim⁑nβ†’βˆž3𝑛2+𝑛)𝑛+𝑛𝑛+𝑛=∞\displaystyle \lim_{n\to \infin} \frac{ 3𝑛^2 + \sqrt𝑛) }{𝑛 + 𝑛\sqrt𝑛 + \sqrt𝑛}=\infin


e.

true

𝑛log𝑛≀n\sqrt𝑛 log 𝑛\le n

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