Calculate the following limit

[ManMan]

This is a limit you're supposed to solve using special limits, I think. I can't do this, no one in my class can do this, so I just assumed it's literally impossible and I brought it here so you can solve it using black magic or something. Thanks, and sorry in case I messed something up.

P.S.: the title of this thread is the exact text of the exercise, it doesn't specify what technique you should use exactly

 

[Dr.Peterson]

This is a limit you're supposed to solve using special limits, I think. I can't do this, no one in my class can do this, so I just assumed it's literally impossible and I brought it here so you can solve it using black magic or something. Thanks, and sorry in case I messed something up.

P.S.: the title of this thread is the exact text of the exercise, it doesn't specify what technique you should use exactly

View attachment 23535

Can you define what you mean by "special limits"?

Have you learned techniques like L'Hopital's rule? What else have you learned about limits?

What is the context in your class? (For example, was this assigned in a way that suggests you're expected to be able to solve it, or as a test of magical abilities?)

Does [MATH][e^{-1}][/MATH] represent how many points you get, or the expected answer, or something else?

 

[Romsek]

Does [MATH][e^{-1}][/MATH] represent how many points you get, or the expected answer, or something else?

[MATH]e^{-1}[/MATH] is the answer

It seems fairly clear that you need to mangle the original expression into a form equivalent to

[MATH]\dfrac{1}{\left(1+\frac 1 n\right)^n}[/MATH]
note that
[MATH]x \to 0 \Rightarrow \sin(x) \approx x \\ x \to 0 \Rightarrow \cos(x) \approx 1[/MATH]

 

[ManMan]

I apologise, I'm Italian and I don't know how to translate the term exactly. I'll try to attach an image of what I mean by special limits


No, I don't know l'Hopital's rule. I know (I'm supposed to know) what a limit is, how a limit looks like when X approaches infinity or a finite value, how to solve limits that turn into [MATH]0/0[/MATH] and such. As for the class, it's a high school class, and I am expected to be able to solve it. I was being ironic with the black magic thing.

 

[ManMan]

[MATH]e^{-1}[/MATH] is the answer

It seems fairly clear that you need to mangle the original expression into a form equivalent to

[MATH]\dfrac{1}{\left(1+\frac 1 n\right)^n}[/MATH]
note that
[MATH]x \to 0 \Rightarrow \sin(x) \approx x \\ x \to 0 \Rightarrow \cos(x) \approx 1[/MATH]

Yes, I tried to do that but failed miserably. I can post an image of my attempt if you want, but it's mostly unintelligible

 

[pka]

This is a limit you're supposed to solve using special limits, I think. I can't do this, no one in my class can do this, so I just assumed it's literally impossible and I brought it here so you can solve it using black magic or something. Thanks, and sorry in case I messed something up.
P.S.: the title of this thread is the exact text of the exercise, it doesn't specify what technique you should use exactly
View attachment 23535

The following two very special and related limits.
\(\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{1}{x}} \right)^x}~\&~\mathop {\lim }\limits_{x \to 0} {\left( {1 - x} \right)^{\frac{1}{x}}}\).
Have you studied one or both of these? The first is the famous exponential and the second is related.
Look these up in your textbook.

 

[Dr.Peterson]

I apologise, I'm Italian and I don't know how to translate the term exactly. I'll try to attach an image of what I mean by special limits
View attachment 23538

No, I don't know l'Hopital's rule. I know (I'm supposed to know) what a limit is, how a limit looks like when X approaches infinity or a finite value, how to solve limits that turn into [MATH]0/0[/MATH] and such. As for the class, it's a high school class, and I am expected to be able to solve it. I was being ironic with the black magic thing.

Don't worry about language; a term like "special" has no unique meaning, and I'd be asking for your list even if you were my neighbor! But your list is appropriate.

And, of course, I, too, was joking about both [MATH]e^{-1}[/MATH] and black magic.

But I don't yet see any good ideas, beyond pka's.

 

[ManMan]

The following two very special and related limits.
\(\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{1}{x}} \right)^x}~\&~\mathop {\lim }\limits_{x \to 0} {\left( {1 - x} \right)^{\frac{1}{x}}}\).
Have you studied one or both of these? The first is the famous exponential and the second is related.
Look these up in your textbook.

Yes, I saw those limits. I've also done other simpler exercises using them. The main problem here is that my chicken brain can't find a way to solve this one specifically. I actually can't solve any of the more complicated ones. I've even looked up videos about this but none of the examples are like this exercise. To be clear, I am aware that you're supposed to turn your weird limit into a cool limit using maths, I even know all the common techniques like multiplying and dividing by the same number, rationalising, that thing where you turn your x into a y... I can't solve a single one of these.

 

[Romsek]

bah... it's clear that the 2nd limit can be directly applied to this problem using the approximations I gave you.

Look deeper.

 

[ManMan]

Is this correct?

 

[Romsek]

using the special limits pka noted you can stop at line 2

[MATH]\lim \limits_{x \to 0} (1-x)^{1/x} = e^{-1}[/MATH]

 

[Dr.Peterson]

using the special limits pka noted you can stop at line 2

[MATH]\lim \limits_{x \to 0} (1-x)^{1/x} = e^{-1}[/MATH]

To me, the first line of the work is informally reasonable, but we need a theorem that says we can just replace sin(x) with x. What theorem is that (specifically, one that would be in a textbook at this level)? Maybe I'm just too accustomed to textbooks that don't cover limits deeply.

@ManMan, do you have such a theorem you can refer to in order to justify your first line?

 

[ManMan]

To me, the first line of the work is informally reasonable, but we need a theorem that says we can just replace sin(x) with x. What theorem is that (specifically, one that would be in a textbook at this level)? Maybe I'm just too accustomed to textbooks that don't cover limits deeply.

@ManMan, do you have such a theorem you can refer to in order to justify your first line?

Not really. I just followed what Romsek said because he wrote "bah" I couldn't disappoint him!!

I tried doing the exercise without using that thing. I don't know if this makes sense. Probably not. I still can't solve most of the exercises lol

Nevermind, I spent fifteen more hours on this one exercise and now it totally works. Here it is.

 

[Steven G]

Does [MATH][e^{-1}][/MATH] represent how many points you get, or the expected answer, or something else?