Proof that <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXAtom-O

Jayla Faulkner

Jayla Faulkner

Answered question

2022-05-15

Proof that lim x x log ( x + 1 x + 10 )
Given this limit:
lim x x log ( x + 1 x + 10 )
I may use this trick:
x + 1 x + 1 = x + 1 x x x + 10
So I will have:
lim x x ( log ( x + 1 x ) + log ( x x + 10 ) ) =
= 1 + lim x x log ( x x + 10 )
But from here I am lost, I still can't make it look like a fondamental limit. How to solve it?

Answer & Explanation

Eden Bradshaw

Eden Bradshaw

Beginner2022-05-16Added 19 answers

lim x x log ( x + 1 x + 10 ) = lim x x log ( 1 9 x + 10 ) = lim x log ( 1 9 x + 10 ) 9 x + 10 ( 9 x x + 10 ) = lim u 0 log ( 1 + u ) u lim x ( 9 x x + 10 ) = 1 ( 9 ) = 9
Marco Villanueva

Marco Villanueva

Beginner2022-05-17Added 6 answers

Putting h = 1 x and assuming Natural Logarithm
lim x x ln x + 1 x + 10 = lim h 0 ln 1 + h 1 + 10 h h
As  ln e a b = ln e a ln e b , ln 1 + h 1 + 10 h = ln ( 1 + h ) ln ( 1 + 10 h )
lim h 0 ln 1 + h 1 + 10 h h = lim h 0 ln ( 1 + h ) ln ( 1 + 10 h ) h
= lim h 0 ln ( 1 + h ) h 10 lim h 0 ln ( 1 + 10 h ) 10 h
Do you know,
lim x 0 ln ( 1 + x ) x = ? ?

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?