Welcome to our community

Be a part of something great, join today!

Limit

Amer

Active member
Mar 1, 2012
275
how to solve this limit

[tex]\displaystyle\lim_{x\rightarrow \infty} \dfrac{\sqrt[x+1]{x+1}-1}{\sqrt[x]{x}-1}[/tex]

(Latex Question :the x of the root is not clear how to make it better )
 
Last edited by a moderator:

chisigma

Well-known member
Feb 13, 2012
1,704
how to solve this limit

[tex]\displaystyle\lim_{x\rightarrow \infty} \dfrac{\sqrt[x+1]{x+1}-1}{\sqrt[x]{x}-1}[/tex]

(Latex Question :the x of the root is not clear how to make it better )
May be that the most comfortable solution is to write...

$\displaystyle \sqrt[1+x]{1+x}= e^{-\frac{\ln (1+x)}{1+x}}$

$\displaystyle \sqrt[x]{x}= e^{- \frac{\ln x}{x}}$

...and then apply l'Hopital's rule...

Kind regards

$\chi$ $\sigma$
 

Amer

Active member
Mar 1, 2012
275
Thanks
 

checkittwice

Member
Apr 3, 2012
37
how to solve this limit

[tex]\displaystyle\lim_{x\rightarrow \infty} \dfrac{\sqrt[x+1]{x+1}-1}{\sqrt[x]{x}-1}[/tex]
Alternative:

You can multiply each of the four terms by [tex]\dfrac{1}{\sqrt[x + 1]{x + 1}}[/tex]