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(Solved): (a) Let f:Z+Z+, with f(n)=4n2+2n+7. Show that f(n)O(n2) : (i) from first principles; (ii) by ...



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(a) Let , with . Show that : (i) from first principles; (ii) by evaluating a suitable limit. [3] (b) For each , define a function : (i) Show that for all . [2] (ii) Give an example of positive integers and with , such that . Briefly justify your answer. (iii) By considering the case when and , explain why Theorem 4.5 in the lecture notes cannot in general give us any information about the asymptotic relationship between and for . [2] (c) For , with , and , let , and let and be two distinct vect or norms. Decide which, if any, of the following functions from to are valid vect or norms. Justify your answers.


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Answer:

(a) (i) To show that    from first principles, we need to find positive constants    such that    for all   

Let's analyze the function   
  

For    we have:
  

Choosing    we can see that    for all   
Therefore,   


please go through to the steps


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