Am I allowed to apply L'Hospital's Rule inside of the natural logarithm function? I have the following limit: lim_(x->oo)ln((2x^2+1)/(x^2+1)) If I was finding the limit of only the terms inside the natural log function, I would have the indeterminate form: oo/oo I want to know if I am allowed to apply L'Hospital's Rule to only the inside terms while ignoring the natural logarithm function, giving me the answer: ln((4x)/(2x))=ln(2)

Brynn Collins

Brynn Collins

Open question

2022-08-20

Am I allowed to apply L'Hospital's Rule inside of the natural logarithm function?
I have the following limit:
lim x ln ( 2 x 2 + 1 x 2 + 1 )
If I was finding the limit of only the terms inside the natural log function, I would have the indeterminate form:

I want to know if I am allowed to apply L'Hospital's Rule to only the inside terms while ignoring the natural logarithm function, giving me the answer:
ln ( 4 x 2 x ) = ln ( 2 )

Answer & Explanation

Kaeden Bishop

Kaeden Bishop

Beginner2022-08-21Added 15 answers

For a continuous function f on a R ¯ we have
lim x a f ( g ( x ) ) = f ( lim x a g ( x ) )
Colton Gregory

Colton Gregory

Beginner2022-08-22Added 4 answers

Yes, you can because
lim x f ( g ( x ) ) = f ( lim x g ( x ) )
But L'Hospital's is not necessary for this case as you can just factor out an x 2 from the numerator and denominator:
lim x ln ( 2 x 2 + 1 x 2 + 1 ) = lim x ln ( 2 + 1 x 2 1 + 1 x 2 )

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