Is it possible to find the limit of (1+1/x)^x as x approaches -infinity?

[manjuvenamma]

Is it possible to find the limit of (1+1/x)^x as x approaches minus infinity using only the fact that it is e if x approaches infinity?

 

[Niladri Dan]

1/e??

 

[manjuvenamma]

1/e??

No, the answer is e. We have to find a reason without using LHospital's rule.

 

[fresh_42]

No, the answer is e. We have to find a reason without using LHospital's rule.

What happens, if you write ##y=-x## and consider ##y \to +\infty##?

 

[manjuvenamma]

I thought about it but got stuck up. Then it becomes (1-1/y)^-y as y approaches infinity which is not a standard formula.

 

[fresh_42]

I thought about it but got stuck up. Then it becomes (1-1/y)^-y as y approaches infinity which is not a standard formula.

Maybe, but ##a^{-y}=(\frac{1}{a})^y## is.

 

[manjuvenamma]

Maybe, but ##a^{-y}=(\frac{1}{a})^y## is.

but 1+1/y is different from 1-1/y. That is where I am stuck.

 

[fresh_42]

but 1+1/y is different from 1-1/y. That is where I am stuck.

For ##y \to \infty##? Isn't it only the difference between left and right?

 

[dgambh]

I thought about it but got stuck up. Then it becomes (1-1/y)^-y as y approaches infinity which is not a standard formula.

I am not sure if this helps but what if you multiply by the conjugate and then try to think why ##\lim_{y \to \infty} \left(1 - \frac{1}{y^2}\right)^y## should be 1.

 

[manjuvenamma]

A great idea but I am not able to think of a proof that it is 1. If this is proved, then I agree that the original question is proved. But how to prove that? It raises other similar interesting questions like what about (1-1/y^3)^y, (1-1/y^4)^(y^2) etc.

 

[Aryamaan Thakur]

I think this problem can be solved in this way. Here is what I think:-

let x = -1/y
then, Lt (1+1/x)^x where x→ -∞
=> Lt (1 - y)^-1/y where y→ 0
Now let -y = 1/p
then, Lt (1 - y)^-1/y where y→ 0
=> Lt (1+1/p)^p where p→∞
And this is equal to e.

 

[ehild]

Is it possible to find the limit of (1+1/x)^x as x approaches minus infinity using only the fact that it is e if x approaches infinity?

You have
[tex]A=\lim _{x\rightarrow\infty} (1+1/x)^x=e[/tex]
[tex]B=\lim _{x\rightarrow\infty} \left(\frac{1}{1-1/x}\right)^x[/tex]
[tex]AB=\lim _{x\rightarrow\infty}\left(\frac{1+1/x}{1-1/x}\right)^x=\lim _{x\rightarrow\infty}(1+2/x)^x=\lim _{x\rightarrow\infty}\left((1+2/x)^{x/2}\right)^2=e^2[/tex]

 

[manjuvenamma]

I am not able to figure out how 1+2/x came. Is it an approx?

 

[ehild]

I am not able to figure out how 1+2/x came. Is it an approx?

Yes, it is an approximation. 1/(1-a) ≈ 1+a and (1+a)²≈1+2a for |a|<<1 . As x-->∞, 1/x -->0 and you can substitute 1/(1-1/x) with (1+1/x). The second approximation is not needed as you have (1+1/x)^x already.
1/(1-a) is the sum of the geometric series 1+a+a²+... . If a <<1, the higher order terms can be neglected.

 

[manjuvenamma]

@ ehild, Great explanation. Thanks to all the respondents who took time to clarify things to me. I would like to know how to write math equations here, is there a link? Is there a way I can accept the answer and close as the raiser of this question?

 

[ehild]

@ ehild, Great explanation. Thanks to all the respondents who took time to clarify things to me. I would like to know how to write math equations here, is there a link? Is there a way I can accept the answer and close as the raiser of this question?

To write nice equations use LaTeX : https://www.physicsforums.com/help/latexhelp/
There are useful symbols also under the ∑ key, and you can make subscript and superscript with the x₂ and x² keys.
I think, closing a thread is possible in the Homework Forum only.

 

[jack action]

This question has been intriguing me, so I've gave it a shot:
$$\left(1+\frac{1}{x}\right)^x = \left(\frac{1+x}{x}\right)^x$$
With ##\infty##:
$$\left(\frac{1+\infty}{\infty}\right)^{\infty}$$
With ##-\infty##:
$$\left(\frac{1-\infty}{-\infty}\right)^{-\infty} = \left(\frac{\infty - 1}{\infty}\right)^{-\infty} = \left(\frac{\infty}{\infty - 1}\right)^{\infty}$$
And one can see that:
$$\frac{\infty}{\infty - 1} = \frac{1+\infty}{\infty}$$
Where the numerator is just 1 + the denominator (like ##\frac{101}{100}## or ##\frac{1000}{999}##). Thus it should give the same answer when at infinity.

 

[fresh_42]

This question has been intriguing me, so I've gave it a shot:
$$\left(1+\frac{1}{x}\right)^x = \left(\frac{1+x}{x}\right)^x$$
With ##\infty##:
$$\left(\frac{1+\infty}{\infty}\right)^{\infty}$$
With ##-\infty##:
$$\left(\frac{1-\infty}{-\infty}\right)^{-\infty} = \left(\frac{\infty - 1}{\infty}\right)^{-\infty} = \left(\frac{\infty}{\infty - 1}\right)^{\infty}$$
And one can see that:
$$\frac{\infty}{\infty - 1} = \frac{1+\infty}{\infty}$$
Where the numerator is just 1 + the denominator (like ##\frac{101}{100}## or ##\frac{1000}{999}##). Thus it should give the same answer when at infinity.

This shouldn't get on my desk for correction.

 

[Charles Link]

This shouldn't get on my desk for correction.

@fresh_42 I agree. They could do a better job of solving this. Suggestion in computing ## y=## limit ## (1+\frac{1}{x})^x ## in a couple of the above posts is to take the natural log of this expression, (so that you have ## ln(y)=x \, ln(1+\frac{1}{x}) ##), use a Taylor expansion of ## ln(1+\frac{1}{x}) ##, (in the form of ## ln(1+u ) ## where ## u ## is small), and then take ## e^{ln(y )}=y ##.

 

[pwsnafu]

Here's a direct method
##\lim_{x\to-\infty}\left(1+\frac{1}{x}\right)^x = \lim_{y\to\infty}\left(1-\frac{1}{y}\right)^{-y} = \lim_{y\to\infty}\frac{1}{\left(1-\frac{1}{y}\right)^y}= \lim_{y\to\infty}\left(\frac{y}{y-1}\right)^{y}.##
Now let ##w = y-1##. Clearly ##y\to\infty## is equivalent to ##w\to\infty##. So
##\lim_{w\to\infty}\left(\frac{w+1}{w}\right)^{w+1}= \lim_{w\to\infty}\left[ \left(1+\frac{1}{w}\right)^w\left(1+\frac{1}{w}\right) \right] = e\times 1 = e## as required.

 

[hunt_mat]

Think about a "subsequence" of the function you've mentioned when [itex]x=n[/itex] an integer. You know that the main sequence and subsequence have the same limit.

 

[Mark44]

This question has been intriguing me, so I've gave it a shot:
$$\left(1+\frac{1}{x}\right)^x = \left(\frac{1+x}{x}\right)^x$$
With ##\infty##:
$$\left(\frac{1+\infty}{\infty}\right)^{\infty}$$
With ##-\infty##:
$$\left(\frac{1-\infty}{-\infty}\right)^{-\infty} = \left(\frac{\infty - 1}{\infty}\right)^{-\infty} = \left(\frac{\infty}{\infty - 1}\right)^{\infty}$$
And one can see that:
$$\frac{\infty}{\infty - 1} = \frac{1+\infty}{\infty}$$
Where the numerator is just 1 + the denominator (like ##\frac{101}{100}## or ##\frac{1000}{999}##). Thus it should give the same answer when at infinity.

It's not legitimate to treat ##\infty## as if it were an ordinary real number.

This shouldn't get on my desk for correction.

? Not sure what you mean here.

 

[Buffu]

Here's a direct method
##\lim_{x\to-\infty}\left(1+\frac{1}{x}\right)^x = \lim_{y\to\infty}\left(1-\frac{1}{y}\right)^{-y} = \lim_{y\to\infty}\frac{1}{\left(1-\frac{1}{y}\right)^y}= \lim_{y\to\infty}\left(\frac{y}{y-1}\right)^{y}.##
Now let ##w = y-1##. Clearly ##y\to\infty## is equivalent to ##w\to\infty##. So
##\lim_{w\to\infty}\left(\frac{w+1}{w}\right)^{w+1}= \lim_{w\to\infty}\left[ \left(1+\frac{1}{w}\right)^w\left(1+\frac{1}{w}\right) \right] = e\times 1 = e## as required.

##\displaystyle \lim_{x\to-\infty}\left(1+\frac{1}{x}\right)^x = \lim_{y\to\infty}\left(1-\frac{1}{y}\right)^{-y} = \left(\lim_{y\to\infty}\left(1-\frac{1}{y}\right)^y\right)^{-1} = (e^{-1})^{-1} = e##

Is this correct ?

 

[fresh_42]

? Not sure what you mean here.

This:

It's not legitimate to treat ##\infty## as if it were an ordinary real number.

Even in cases where such considerations might lead to correct results, I find it a bad habit to use them - far too dangerous w.r.t. possible errors. And by no means this should be taught, IMO. Maybe it can be used to get roughly a picture of the situation, but even this can already be misleading.

 

[jack action]

It's not legitimate to treat ##\infty## as if it were an ordinary real number.

FWIW, I did not treat ##\infty## as a real number, otherwise the following wouldn't be true:
$$\frac{\infty}{\infty - 1} = \frac{1+\infty}{\infty}$$
But I admit that it would've looked a lot more professional by using ##x## and ##\displaystyle \lim_{x \rightarrow \infty}##. That is why I liked post #20 which essentially demonstrate the same thing I did. So this should be better:
$$\lim_{x \rightarrow \infty}\frac{x}{x - 1} = \lim_{x \rightarrow \infty}\frac{1+x}{x}$$

 

[RogueOne]

Is it possible to find the limit of (1+1/x)^x as x approaches minus infinity using only the fact that it is e if x approaches infinity?

Try to graph this:

[1-(1/(-x))]^(-x)

Understanding that the shaded region in the below graph, for the equation "y= 1/x", is equal to 1 square unit.