L'Hospital of e^(1/x) as limit x-> 0- and 0+

[thrust_1]

e^(1/x) as limit x-> 0- and 0+

Homework Statement

lt e^(1/x) = 0
x->0-

and

lt e^(1/x) = +( infinity)
x->0+

Homework Equations

The Attempt at a Solution

let y = e^(1/x)
ln y = (1/x) ln e

lt (ln e)/x
x->0-

applying L'Hospitals rule = 0

How to procedd with the lt x-> 0+

Kindly help me.

 

[gb7nash]

You're not using l'hopital's rule correctly. To use l'hopital, you need +-inf/inf, or 0/0, neither of which you have. It doesn't look like you're even taking the derivatives? You're using a different technique.

In any case, why are you trying to use l'hopital's rule for this? You can evaluate it directly. For:

\lim_{x \to 0^{-}}e^{\frac{1}{x}}

What is the exponent approaching? What is e raised to this? Do the same thing for the other problem.

 

[thrust_1]

The question was just to prove the limit. So that's why i used L'hospitals rule.

 

[gb7nash]

The question was just to prove the limit. So that's why i used L'hospitals rule.

Using l'hopital's rule is no more legit than evaluating the limit directly. If you want to prove the limit, you need to use the epsilon-delta definition of limits.

 

[thrust_1]

Thanks for the explanation

 

[vela]

Using l'hopital's rule is no more legit than evaluating the limit directly. If you want to prove the limit, you need to use the epsilon-delta definition of limits.

I wouldn't say you have to use the delta-epsilon definition. Evaluating the limit is fine, but you just need to justify each step.

The question was just to prove the limit. So that's why i used L'hospitals rule.

While you can use L'Hopital's rule as part of a proof, you can only do so when it's applicable to the particular problem, which it isn't in this case.

 

[redargon]

Also, it is L'Hopital's rule not L'Hospital's rule. That way people won't look at you funny when you asked for help on the problem

 

[HallsofIvy]

Also, it is L'Hopital's rule not L'Hospital's rule. That way people won't look at you funny when you asked for help on the problem

Wolfram Mathworld http://mathworld.wolfram.com/LHospitalsRule.html:
"Note that l'Hospital's name is commonly seen spelled both "l'Hospital" (e.g., Maurer 1981, p. 426; Arfken 1985, p. 310) and "l'Hôpital" (e.g., Maurer 1981, p. 426; Gray 1997, p. 529), the two being equivalent in French spelling."

 

[gb7nash]

I wouldn't say you have to use the delta-epsilon definition. Evaluating the limit is fine, but you just need to justify each step.

It depends how technical you want to be. I can eyeball something like:

\lim_{x \to 0} x^2

and tell you that the limit is 0, or I can simply plug in values that get closer and closer to 0. Did I prove it? No. However, just about anyone can see that the limit is 0. If you want to actually prove it mathematically, the only way I can see this possible is by using e-d. If I'm overloking something though, I'm all ears.

edit:

I think I may know what you're talking about. For a lot of problems, you can take exploit continuity. In my example (and a lot of examples), you can. This of course assumes that you're allowed to assume the function is continuous at the point in question (which requires e-d and evaluating the limit there anyways). However, whenever I see "prove the limit" in a calculus book, I never think of evaluating it.

 

[redargon]

Wolfram Mathworld http://mathworld.wolfram.com/LHospitalsRule.html:
"Note that l'Hospital's name is commonly seen spelled both "l'Hospital" (e.g., Maurer 1981, p. 426; Arfken 1985, p. 310) and "l'Hôpital" (e.g., Maurer 1981, p. 426; Gray 1997, p. 529), the two being equivalent in French spelling."

I agree fully, but the rules for french spelling have changed.

"In the 17th and 18th centuries, the name was commonly spelled "l'Hospital", however, French spellings have been altered: the silent 's' has been dropped and replaced with the circumflex over the preceding vowel." - http://en.wikipedia.org/wiki/Guillaume_de_l'Hôpital

Similar to forêt and forest... I actually wonder if the name was changed posthumously or if he started life as l'Hôpital, any ideas?

Either way, the pronounciation should definitely not include the 's'. That's why I mentioned it, so that the OP doesn't go around asking colleagues or professors about the l'Hospital rule using the english pronounciation for hospital, which may cause confusion or get a giggle out of said colleague or professor