# Limit

#### Amer

##### Active member
how to solve this limit

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

(Latex Question :the x of the root is not clear how to make it better )

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#### chisigma

##### Well-known member
how to solve this limit

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

(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

$\chi$ $\sigma$

Thanks

#### checkittwice

##### Member
how to solve this limit

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

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