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March 4th, 2019, 11:18 AM  #1 
Newbie Joined: Mar 2019 From: Romania Posts: 1 Thanks: 0  Need some help to solve this limit
Hi everyone, I have a problem that I don't know how to solve  ideally without using l'Hospital. Can you pls help? 
March 5th, 2019, 06:38 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2666 Math Focus: Mainly analysis and algebra 
First, writing $y=\frac1x$ we can see that there are actually two problems here: $$\lim_{x \to 0^} \frac1x e^{\frac1x} = \lim_{y \to \infty} y e^{y} \to (\infty) \cdot \infty$$ So the function grows without bound in the negative direction. The second is $$\lim_{x \to 0^+} \frac1x e^{\frac1x} = \lim_{y \to +\infty} y e^{y} = \lim_{y \to +\infty} \frac{y}{ e^{y}} = 0$$ This is a standard result which you would normally be allowed to quote directly. Since there are two different values for these two limits, you should be able to determine the correct answer to the given problem. Alternatively, you could graph the function and come to the same conclusion. Last edited by v8archie; March 5th, 2019 at 06:42 AM. 
March 5th, 2019, 01:04 PM  #3  
Global Moderator Joined: May 2007 Posts: 6,823 Thanks: 723  Quote:
 

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