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  • Evaluate   <img alt="\mathrm{\lim _{x \rightarrow+\infty}\left(\frac{(\ln (x+1))^x}{(2 x)^{x / 3}}\right)}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim%20_%7Bx

Evaluate   \mathrm{\lim _{x \rightarrow+\infty}\left(\frac{(\ln (x+1))^x}{(2 x)^{x / 3}}\right)}

Option: 1

0


Option: 2

1


Option: 3

2


Option: 4

5


Answers (1)

best_answer

we will use the fact that   \mathrm{\ln (x+a)=\ln x+\frac{a}{x}+\cdots}  for  x large.

                                                                let  \mathrm{y=\frac{(\ln (x+1))^x}{(2 x)^{x / 3}}}.

then

                          \mathrm{\begin{aligned} \ln y & =x \ln (\ln (x+1))-\frac{x}{3}(\ln 2+\ln x) \\\\ & =x \ln \left(\ln x+\frac{1}{x}+\cdots\right)-\frac{x}{3}(\ln 2+\ln x) \\\\ & =x\left(\ln (\ln x)+\frac{1}{x \ln x}+\cdots\right)-\frac{x}{3}(\ln 2+\ln x) \\\\ & =-\frac{x}{3} \ln x+\cdots \rightarrow-\infty \end{aligned}}

therefore

                                    \mathrm{\lim _{x \rightarrow+\infty}\left(\frac{(\ln (x+1))^x}{(2 x)^{x / 3}}\right)=0}.

Posted by

SANGALDEEP SINGH

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