Find the limit or explain why it diverges?

Question

Find the limit of #sqrt(9n^2-5n+3)/root(4)(16n^4-7n^3+4n+1)#

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Answer 1 · 2017-07-20T08:54:18

#3/2#

#sqrt(9n^2-5n+3)/root(4)(16n^4-7n^3+4n+1) = n/n sqrt(9-5/n+3/n^2)/root(4)(16-7/n+4/n^3+1/n^4) = sqrt(9-5/n+3/n^2)/root(4)(16-7/n+4/n^3+1/n^4)#

now

#lim_(n->oo)sqrt(9n^2-5n+3)/root(4)(16n^4-7n^3+4n+1) =lim_(n->oo)sqrt(9-5/n+3/n^2)/root(4)(16-7/n+4/n^3+1/n^4) =#

#=(lim_(n->oo)sqrt(9-5/n+3/n^2))/(lim_(n->oo)root(4)(16-7/n+4/n^3+1/n^4)) = sqrt(9)/root(4)(16) = 3/2#