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# What is Lim x-> infinity of (e^x + x)^1/x

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

Guest Mar 25, 2015

#2
+87301
+10

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
+92781
+10

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
+87301
+10

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
+92781
+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