# Lim x→∞ (x[(1 + 1/x)^x] - e)

how are u every one there
i am happy to be one of ur friends
just we have challenge because of this limit
so i hope that someone donate to solve it
thanx

I would suggest changing this in a 0/0 or an $$\infty / \infty$$ and applying L' Hopital.

Last edited:
But it looks obvious that the limit does not exist !!!

Ow yes, you're right. It obviously doesnt exist. I read one of the parantheses wrong.

HallsofIvy
Homework Helper
Kahlan, the "wrong parentheses" micromass was seeing was
$$\lim_{x\to\infty}x[(1+ 1/x)^x- e]$$

It is well known that $(1+ 1/x)^x$ goes to e so that (1+ 1/x)^x- e would go to 0- the additional x outside the braces would give an indeterminant form of "infinity* 0".

However, you have the "x" inside the braces and -e outside. $\lim_{x\to\infty}x(1+ 1/x)^x$ is of the form "infinity*e" which does not converge.

$$\lim_{x\to\infty}x[(1+ 1/x)^x- e]$$

it sould be like $$\lim_{x\to\infty}x[(1+ 1/x)^x- e]$$
sorry guys
as well as answer will be 1

how are u every one there
i am happy to be one of ur friends
just we have challenge because of this limit
so i hope that someone donate to solve it
thanx

sorry

The limit does exist.

You have:
x[(1 + 1/x)^x] - e = x[exp(x*ln(1+1/x))-e)
Then you use h=1/x, what gives:

1/h*[exp(ln(1+h)/h)-e)
=1/h*[exp(1/h*(h-h²/2+h^3/3+o(h))-e)]
=1/h*[exp(1-h/2+h²/3+o(h))-e)]
=1/h*[e(exp(-h/2+h²/3+o(h))-1)]
=1/h*[e(1-h/2+h²/3+o(h)-1)
=1/h*[e(-h/2+h²/3+o(h)]
=e/h*(-h/2+h²/3+o(h)]
=-e/2+h/3+o(h)

As x→∞, then h→0

Finally, you get:

lim x→∞ (x[(1 + 1/x)^x] - e) = lim h→0 (-e/2+h/3+o(h)) = -e/2

D H
Staff Emeritus
scichem, you're being a bit sloppy with braces and parentheses, so it's hard to see where you went wrong. You should have obtained

$$x\bigl((1 + 1/x)^x - e\bigr) = -\frac e 2 + \frac{11 e}{24}\frac 1 x - O\bigl(1/x^2\bigr)$$
or
$$\bigl((1 + h)^{1/h} - e\bigr)/h = -\frac e 2 + \frac{11 e}{24}h - O\bigl(h^2\bigr)$$

In the limit x→∞ or h→0, these become -e/2.

Somehow you did obtain the right limit.

scichem, you're being a bit sloppy with braces and parentheses, so it's hard to see where you went wrong. You should have obtained

$$x\bigl((1 + 1/x)^x - e\bigr) = -\frac e 2 + \frac{11 e}{24}\frac 1 x - O\bigl(1/x^2\bigr)$$
or
$$\bigl((1 + h)^{1/h} - e\bigr)/h = -\frac e 2 + \frac{11 e}{24}h - O\bigl(h^2\bigr)$$

In the limit x→∞ or h→0, these become -e/2.

Somehow you did obtain the right limit.

You're right, thanks for noticing, I've forgotten something in the series expansion of exp(u), actually the u²/2 part with u=-h/2+h²/3+o(h²), which actually gives the element h²/8:

1/h*[exp(ln(1+h)/h)-e)
=1/h*[exp(1/h*(h-h²/2+h^3/3+o(h²))-e)]
=1/h*[exp(1-h/2+h²/3+o(h²))-e)]
=1/h*[e(exp(-h/2+h²/3+o(h²))-1)]
=1/h*[e(1-h/2+h²/3+o(h²)-1)
=1/h*[e(-h/2+h²/3+o(h²)]
=e/h*(-h/2+h²/3+h²/8+o(h²)]
=e/h*(-h/2+11h²/24+o(h²)

=-e/2+11h/24+o(h)

And then you find the limit -e/2.

PS: Where can I find a tutorial/help section on this forum to know how to write series expansion formulas like what you've done?

D H
Staff Emeritus
PS: Where can I find a tutorial/help section on this forum to know how to write series expansion formulas like what you've done?