How to find the limit of the sequence with n-th

Sullivan Pearson

Sullivan Pearson

Answered question

2022-04-22

How to find the limit of the sequence with n-th term
xn=(2n+3n2)n

Answer & Explanation

2sze1c1se3nh

2sze1c1se3nh

Beginner2022-04-23Added 17 answers

As usual when the variable appears in the exponent, write it using the rule ab=exp(blna)
exp(nln(2n+3n2))
and
nln(2n+3n2=nln(2n+3)2nln(n)
=nln(n)(ln(2+3n)2)
It is now easy to conclude: nln(n) and ln(2+3n)2ln22<0, so nln(2n+3n2) and exp(nln(2n+3n2))0
To have a chance to apply l’Hospital rule, try to put x=1n and study the limit in 0, but I think this is not very useful here, as the above method is rather straightforward.

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