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

Evaluate   \mathrm{\lim _{x \rightarrow \infty} x\left(\left(1+\frac{1}{x}\right)^x-e\right)}

Option: 1

-1/2


Option: 2

3/5


Option: 3

1


Option: 4

0


Answers (1)

best_answer

The idea of doing  \mathrm{x=1 / t}  is good:

                                  \mathrm{\lim _{t \rightarrow 0^{+}} \frac{(1+t)^{1 / t}-e}{t}}

If  \mathrm{f(t)=(1+t)^{1 / t}, \text { then, for } t>0},

                                     \mathrm{f^{\prime}(t)=(1+t)^{1 / t} \frac{t-(1+t) \log (1+t)}{t^2(1+t)}}

Since  \mathrm{\lim _{t \rightarrow 0^{+}} \frac{(1+t)^{1 / t}}{1+t}=e}, you are reduced to computing 

                                 \mathrm{\lim _{t \rightarrow 0^{+}} \frac{t-(1+t) \log (1+t)}{t^2}=\lim _{t \rightarrow 0^{+}} \frac{1-1-\log (1+t)}{2 t}=-\frac{1}{2}}

Posted by

Ritika Jonwal

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