How to solve <munder> <mo form="prefix">lim <mrow class="MJX-TeXAtom-ORD"> n

Petrovcic2x

Petrovcic2x

Answered question

2022-06-22

How to solve lim n ( a 1 n + b 1 n 2 ) n

Answer & Explanation

plodno8n

plodno8n

Beginner2022-06-23Added 17 answers

Assuming a , b > 0
We have
( a 1 n + b 1 n 2 ) n = a ( 1 + r 1 n 2 ) n ,
where r = b / a. We can write
a ( 1 + r 1 n 2 ) n = a ( 1 1 r 1 n 2 ) n = a exp [ n log ( 1 1 r 1 n 2 ) ] = a exp [ n log ( 1 1 e 1 n log r 2 ) ] = a exp [ n log ( 1 1 n log r + o ( 1 n 2 ) 2 ) ] = a exp [ n log ( 1 + 1 2 n log r + o ( 1 n 2 ) ) ] = a exp [ 1 2 log r + o ( 1 n ) ) ] a e 1 2 log r = a r 1 / 2 = a b .

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