L'Hôpital's as x tends to infinityI'm searching for the explanation to the limit of: lim...

Livia Cardenas

Livia Cardenas

Answered

2022-07-22

L'Hôpital's as x tends to infinity
I'm searching for the explanation to the limit of:
lim x x ln x + 1 x 1 .
I know the answer is 2, but I can't seem to get there. The problem is in my textbook under a section with l'Hôpital.

Answer & Explanation

Kitamiliseakekw

Kitamiliseakekw

Expert

2022-07-23Added 23 answers

Hint:
ln x + 1 x 1 = ln ( x + 1 ) ln ( x 1 )
so that
x ln x + 1 x 1
can be rewritten as
ln ( x + 1 ) ln ( x 1 ) x 1 .
Now,
lim x x ln x + 1 x 1 = lim x 1 x + 1 1 x 1 1 x 2
Find a common denominator for the two terms in the numerator, cancel fractions, and you will arrive at the desired answer of 2.
Note: Indeed, ln x + 1 x 1 0 and x 1 0 as x . Therefore, we have the required form 0 / 0 and L'Hopital's rule is applicable.

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