A limit problem without the use of a Taylor series expansion

Homework Statement:

This is a famous problem of limit.
Lim ((1+x)^(1/x)-e+ex/2)/x^2 where x tends to 0.

We can easily solve it with taylor expansion but i want to solve it without using taylor expansion.Is there a way?

Relevant Equations:

Taylor expansion
I tried substituting x=cos2theta but it was of no use.I thought many ways but i could not make 0/0 form.So please help.

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fresh_42
Mentor
Can you type this in LaTeX so that the function can be read more easily? See https://www.physicsforums.com/help/latexhelp/ for how it is done. I have read ##\lim_{x\to 0} \dfrac{(1+x)^{\frac{1}{x}}-e-e\cdot \frac{x}{2}}{x^2}## but I'm not sure. Have you tried L'Hôpital?

Can you type this in LaTeX so that the function can be read more easily? See https://www.physicsforums.com/help/latexhelp/ for how it is done. I have read ##\lim_{x\to 0} \dfrac{(1+x)^{\frac{1}{x}}-e-e\cdot \frac{x}{2}}{x^2}## but I'm not sure. Have you tried L'Hôpital?
Yes you have taken it correct.I think we cannot apply L' Hospital since 1^infinity is there and therefore no 0/0 or infinity/infinity form.

fresh_42
Mentor
And why isn't it simply ##-\infty##?
##\lim_{x\to 0} \dfrac{(1+x)^{\frac{1}{x}}-e-e\cdot \frac{x}{2}}{x^2}=
\lim_{x\to 0} \dfrac{e-e-e\cdot \frac{x}{2}}{x^2} =
\lim_{x\to 0} \dfrac{-e}{2x}##

And why isn't it simply ##-\infty##?
##\lim_{x\to 0} \dfrac{(1+x)^{\frac{1}{x}}-e-e\cdot \frac{x}{2}}{x^2}=
\lim_{x\to 0} \dfrac{e-e-e\cdot \frac{x}{2}}{x^2} =
\lim_{x\to 0} \dfrac{-e}{2x}##
So will L'Hospital work here.

fresh_42
Mentor
I haven't used L'Hôpital, only that ##\lim_{x \to 0} (x+1)^{\frac{1}{x}} =e##. I think that is a bit cheating, as I ignored the denominator, so it has to be shown formally that ##\lim_{x\to 0} \dfrac{(x+1)^{\frac{1}{x}}-e}{x^2}=0## by whatever method.

I tried L'Hospital but it is becoming complicated.

I haven't used L'Hôpital, only that ##\lim_{x \to 0} (x+1)^{\frac{1}{x}} =e##. I think that is a bit cheating, as I ignored the denominator, so it has to be shown formally that ##\lim_{x\to 0} \dfrac{(x+1)^{\frac{1}{x}}-e}{x^2}=0## by whatever method.

fresh_42
Mentor
This is because I made a sign error in post #2 which you confirmed in post #3. Please use LaTeX next time. Seems we cannot simplify ##(1+x)^{1/x}=e## in the limit. I had such a feeling. In this case I would try L'Hôpital, but Taylor is easier.

This is because I made a sign error in post #2 which you confirmed in post #3. Please use LaTeX next time. Seems we cannot simplify ##(1+x)^{1/x}=e## in the limit. I had such a feeling. In this case I would try L'Hôpital, but Taylor is easier.
Ok i will use LaTeX next time.But can you please solve it without Taylor.

fresh_42
Mentor
Ok i will use LaTeX next time.But can you please solve it without Taylor.
This means in other words: Can you prove that the third term of the Taylor series of ##\dfrac{(1+x)^\frac{1}{x}}{x^2}## equals ##\frac{11}{24}\,e## without using the Taylor series? However, already ##f''(0)## is an ugly limit. Another wording is: How fast exactly does ##(1+x)^\frac{1}{x}## converge to ##e##.

I'm sure there is a nice trick to manage those expressions, but I don't know any.

This means in other words: Can you prove that the third term of the Taylor series of ##\dfrac{(1+x)^\frac{1}{x}}{x^2}## equals ##\frac{11}{24}\,e## without using the Taylor series? However, already ##f''(0)## is an ugly limit. Another wording is: How fast exactly does ##(1+x)^\frac{1}{x}## converge to ##e##.

I'm sure there is a nice trick to manage those expressions, but I don't know any.

Homework Helper
The limit is not -11e/24 -- the limit does not exist.

a) You cannot expand the numerator in a series around 0 since the exponential term is not defined for x = 0, so that approach does not apply

b) More directly, a plot of the expression (look at the plot from (say) x = 0.001 to about x = 0.3) shows a vertical asymptote tending to negative infinity as you approach 0 from the right.

fresh_42
Mentor
The limit is not -11e/24 -- the limit does not exist.

a) You cannot expand the numerator in a series around 0 since the exponential term is not defined for x = 0, so that approach does not apply

b) More directly, a plot of the expression (look at the plot from (say) x = 0.001 to about x = 0.3) shows a vertical asymptote tending to negative infinity as you approach 0 from the right.
WolframAlpha says differently:
https://www.wolframalpha.com/input/?i=limit+(x+to+0)+((((1+x)^(1/x))-e+(ex/2))/(x^2))

Homework Helper
I find Alpha's "series expansion" suspicious. The requirement for that to be valid are that the function in question be differentiable about the point in question (0, in this case), and this function is not defined at 0.

Also: it seems that there are two expressions being bantered about here. The original expression was

##
\lim_{x \to 0} \dfrac{\left(1+x\right)^{(1/x)} - e + \frac{ex}2}{x^2}
##

The other is
##
\lim_{x \to 0} \dfrac{\left(1+x\right)^{(1/x)} - e - \frac{ex}2}{x^2}
##

Which one is correct?

fresh_42
Mentor
I find Alpha's "series expansion" suspicious. The requirement for that to be valid are that the function in question be differentiable about the point in question (0, in this case), and this function is not defined at 0.

Also: it seems that there are two expressions being bantered about here. The original expression was

##
\lim_{x \to 0} \dfrac{\left(1+x\right)^{(1/x)} - e + \frac{ex}2}{x^2}
##

The other is
##
\lim_{x \to 0} \dfrac{\left(1+x\right)^{(1/x)} - e - \frac{ex}2}{x^2}
##

Which one is correct?
See post #11.

See post #11.
It is not yet solved.Can you help fresh.

fresh_42
Mentor
No, I have no idea. I tried a few things but they led nowhere. The margin is very, very narrow and one will need many terms of the Taylor expansion of ##(1+x)^{1/x}##. I substituted ##n= 1/x## and worked with natural numbers, but calculation with just ##O(\frac{1}{n})## was not precise enough. Even integration uses the series expansion.

Last edited:
SammyS
Staff Emeritus
Homework Helper
Gold Member
I find Alpha's "series expansion" suspicious. The requirement for that to be valid are that the function in question be differentiable about the point in question (0, in this case), and this function is not defined at 0.

Also: it seems that there are two expressions being bantered about here. The original expression was

##
\lim_{x \to 0} \dfrac{\left(1+x\right)^{(1/x)} - e + \frac{ex}2}{x^2}
##

The other is
##
\lim_{x \to 0} \dfrac{\left(1+x\right)^{(1/x)} - e - \frac{ex}2}{x^2}
##

Which one is correct?
The correct expression is ##\ \displaystyle
\lim_{x \to 0} \dfrac{\left(1+x\right)^{(1/x)} - e + \dfrac{e \cdot x}2}{x^2}
## .

Although the function, ## \left(1+x\right)^{(1/x)} ## is not defined at ## x=0 ##, this is a removable discontinuity.

The following function is defined and differentiable at ##x=0##, and has the Taylor expansion given by WolframAlpha for the function, ## \left(1+x\right)^{(1/x)} ##.

##\displaystyle \quad \quad u(x) = \begin{cases} \left(1+x\right)^{(1/x)} & \text{if } x \ne 0 \\ e & \text{if } x = 0 \end{cases} ##

.

The correct expression is ##\ \displaystyle
\lim_{x \to 0} \dfrac{\left(1+x\right)^{(1/x)} - e + \dfrac{e \cdot x}2}{x^2}
## .

Although the function, ## \left(1+x\right)^{(1/x)} ## is not defined at ## x=0 ##, this is a removable discontinuity.

The following function is defined and differentiable at ##x=0##, and has the Taylor expansion given by WolframAlpha for the function, ## \left(1+x\right)^{(1/x)} ##.

##\displaystyle \quad \quad u(x) = \begin{cases} \left(1+x\right)^{(1/x)} & \text{if } x \ne 0 \\ e & \text{if } x = 0 \end{cases} ##

.
Do you have any idea to solve it without taylor expansion.

SammyS
Staff Emeritus
Homework Helper
Gold Member
Do you have any idea to solve it without Taylor expansion.
I don't have any good idea for that, but I'm thinking ...

As has been pointed out, the derivatives of the numerator get very complicated.

vela
Staff Emeritus