How to find the limit \lim_(x->oo)(ln(x^2+x+5))/(ln(x^8-x+3))

Humberto Campbell

Humberto Campbell

Answered question

2022-11-14

How to find the limit lim x ln ( x 2 + x + 5 ) ln ( x 8 x + 3 ) ?
I tried reducing it to the form lim ( 1 + x n ) 1 / x n = 1, but that didn't work.

Answer & Explanation

artirw9f

artirw9f

Beginner2022-11-15Added 20 answers

Factoring, we get:
x 2 + x + 5 = x 2 ( 1 + 1 / x + 5 / x 2 ) .
Using the above and log rules:
ln ( x 2 + x + 5 ) = ln ( x 2 ) + ln ( 1 + 1 / x + 5 / x 2 ) = 2 ln ( x ) + ln ( 1 + 1 / x + 5 / x 2 ) .
Similarly:
ln ( x 8 x + 3 ) = ln ( x 8 ) + ln ( 1 / x 7 + 3 / x 8 ) = 8 ln ( x ) + ln ( 1 + 1 / x 7 + 3 / x 8 ) .
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