Determine the following limit as x approaches 0: ln &#x2061;<!-- ⁡ -->

Kenley Wagner

Kenley Wagner

Answered question

2022-05-21

Determine the following limit as x approaches 0: ln ( 1 + x ) x
The process I want to take to solving this is by using the definition of the limit, but I am getting confused. ( without l'hopitals rule)
lim h 0 f ( x + h ) f ( x ) h
lim h 0 ln ( 1 + x + h ) x + h ln ( 1 + x ) x h
lim h 0 x ln ( 1 + x + h ) ( x + h ) ln ( 1 + x ) h x ( x + h ) )
At this point I get confused because I know the answer is 1, but I am not getting this answer through simplification of my formula.

Answer & Explanation

Rubi Boyle

Rubi Boyle

Beginner2022-05-22Added 14 answers

You are talking about L'Hôpital's rule, so I assume you already know how to differentiate the logarithm. Now note, that
log ( x + 1 ) x = log ( x + 1 ) log ( 1 ) ( x + 1 ) 1
Thus
lim x 0 log ( x + 1 ) x = lim x 0 log ( x + 1 ) log ( 1 ) ( x + 1 ) 1 = ( log ( x ) ) x = 1 = 1 x | x = 1 = 1
(This is not by using L'Hôpital's rule but only by using the definition of derivative and knowing the derivative of log ( x ))
Mackenzie Rios

Mackenzie Rios

Beginner2022-05-23Added 4 answers

Introduce a new variable u = 1 / x. Then you limit becomes
lim u ln ( 1 + 1 / u ) 1 / u
mulitply numerator and denominator by u, you get
lim u u [ ln ( 1 + 1 / u ) ]
move u into the log, getting
lim u [ ln ( 1 + 1 / u ) u ]
then, since ln is continuous
ln ( lim u ( 1 + 1 / u ) u )
limit inside is equal to e and ln ( e ) = 1

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