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What is Lim x-> infinity of (ex + x)1/x

Guest Mar 25, 2015

Best Answer 

 #2
avatar+80983 
+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
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3+0 Answers

 #1
avatar+91450 
+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 (ex + 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
avatar+80983 
+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
 #3
avatar+91450 
+5

Thanks Chris, I didn't even think to check on a graph. 

You always think of graphs more readily than I do :)

Melody  Mar 26, 2015

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