Finding the Limit of n(ln(n+10) - ln(n))

[physstudent1]

Homework Statement

find the limit as n->oo of n(ln(n+10)-ln(n))

Homework Equations

The Attempt at a Solution

the answer is 10 this is also part of my exam review I am pretty lost on this one I tried using log rules to combine the terms tried getting it in form for L'H but I don't know what to do.

 

[Dick]

What did you try?

 

[physstudent1]

I think I just found the answer...I combined the ln terms so ln((n+10)/n)^n = L then I exponentiated both sides to give ((n+10)/n)^n= e^L then The limit of (1 + 10/n)^n = e^10 = e^L so taking the ln of both sides gives 10 = L (I know the limit of (1 + 10/n)^n = e^10 because it is a given formula on my formula sheet)

 

[Dick]

That's one way. To use log rules try ln(n+10)-ln(n)=ln((n+10)/n)=ln(1+10/n). Now write it as ln(1+10/n)/(1/n) which is 0/0 and do l'Hopital.

 

[physstudent1]

I keep getting the wrong answer with this method I do L'H from where you said and I get (n/(1+10)*(-1(1+10)/n^2)*-n^2 ...

 

[physstudent1]

finally got it thanks

 

[Kurret]

Even if you got it, I can point out another method. Combine the two logarithms to get n*ln(1+10/n), then write ln as an infinite sum (ln(1+x)=x-x^2/2+x^3/3-... if -1<x<1, which obviously is true for n->oo), and the solution comes immediately.

 

[m_s_a]

lim(n(ln(n+10)-lnn)=10
n__>inf

 

[m_s_a]

the lim.solution: