#2**+10 **

Your answer is correct, Melody...look at the graph....https://www.desmos.com/calculator/0wasx7syzk

This approches 2.718..... = 'e" as x approaches infinity....

[Nice trick of multiplying the top and bottom by e^x....!!! ]

CPhill
Mar 26, 2015

#1**+10 **

Please can another mathematician check this answer.

I'm not good at limits so my answer will probably be wrong but I always think I learn better if I get involved.

Lim x-> infinity of (e^{x} + x)^{1/x}

$$\\\displaystyle\lim_{x\rightarrow\infty}\;(e^x+x)^{1/x}\\\\

=\displaystyle\lim_{x\rightarrow\infty}\;\left(\frac{e^x}{1}\cdot \frac{(e^x+x)}{e^x}\right)^{1/x}\\\\

=\displaystyle\lim_{x\rightarrow\infty}\;\left[\left(1+ \frac{x}{e^x}\right)^{1/x}\cdot (e^x)^{1/x}\right]\\\\

=\displaystyle\lim_{x\rightarrow\infty}\;\left[\left(1+ \frac{x}{e^x}\right)^{1/x}\cdot e\right]\\\\

=\;e\times\;\left[\left(1+ 0}\right)^{1/x}\right]\\\\

=\;e$$

Melody
Mar 26, 2015

#2**+10 **

Best Answer

Your answer is correct, Melody...look at the graph....https://www.desmos.com/calculator/0wasx7syzk

This approches 2.718..... = 'e" as x approaches infinity....

[Nice trick of multiplying the top and bottom by e^x....!!! ]

CPhill
Mar 26, 2015