calculus ?

Question

Find the limit of the sequence:
a_n
= (5n^2+5n+6)/(3n^2+7n+5)

Thanks for your help!

Answer 1

lim [ (5n^2 + 5n + 6) / (3n^2 + 7n + 5) ]
n -> infinity

Your first step is to divide each term by the highest power of n; in this case, it's n^2.

lim [ (5 + 5/n + 6/n^2) / (3 + 7/n + 5/n^2)
n -> infinity

Now, evaluate this term by term, remembering that as n goes to infinity, c/n^k approaches 0.

(5 + 0 + 0) / (3 + 0 + 0)

5/3

Answer 2

(5n^2+5n+6)/(3n^2+7n+5)
Apply L'Hospital's rule
(10n+5)/(6n+7)
Divide numerator and denominator by n getting:
(10+5/n)/(6+7/n)
As n --> infinity the limit becomes 10/6 = 5/3

Answer 3

just use l'hospitals' rule
limit n-->infinity (5n^2+5n+6)/(3n^2+7n+5)=
limit n-->infinity (10n +5)/(6n+7)=
limit n-->infinity (10)/(6) =
5/3

Answer 4

First, ignore all the terms less than n², as they become insignificant for sufficiently large values of n (and in general, you may ignore all the terms smaller than the term of greatest degree). So this is lim 5n²/(3n²), which is obviously 5/3.

Answer 5

5/3

Pascal explained it well...

Apply L'Hospital's rule? -- Why use a cannon to kill a butterfly?