Limit

[Amer]

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 )

 

[chisigma]

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]

Thanks

 

[checkittwice]

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]