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# Behavior towards infinity and graph of e^-x+e^(x+e^-x)

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In class we covered behavior of infinity of e equations. I came up with the following equation e^-x+e^(x+e^-x). You can simplify it to e^e^-x so if x →+∞ you approach 1 but if you try a large number for x you get a big number so shouldn't the graph look like this? My teacher said that Iit is going to curve down. Is this true and if yes, can you find out when the graüh does start to approach 1. (Sorry iff my english is bad and if I cant describe the problem well enough, english isn't my first language) Nov 13, 2023

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Firstly I do not think your simplification is correct.  (Unless I have misunderstood what you have wanted to display)

$$e^{-x}+e^{x+e^{-x}}\\ =e^{-x}+e^xe^{e^{-x}}\\ \displaystyle \lim_{x\rightarrow \infty}\;e^{-x}+e^xe^{e^{-x}}\\ =0+e^\infty*1\\ =e^\infty\\ =\infty$$

My presentation is not great. Limits is not my strong suit.  But still I think the logic is correct.

Here is the graph

https://www.desmos.com/calculator/xmejid73ie

Nov 13, 2023