# Calculate the limit

LCSphysicist
Homework Statement:
.
Relevant Equations:
(1 + a^x)^(a^(-x))
(1 + a^x)^(a^(-x))
Let's assume a, say, two.

the limit of it, with x tending to infinity, is one, but i was thinking...
Calling 2^x by a, we have that when x tend to infinity, so do a, So: that is euler number no? Contradictory... where am i wrong?

#### Attachments

Eclair_de_XII
one

That's what I got. The way I usually deal with limits of this form is to express it as a function of the exponential function and the natural logarithm, then use l'Hopital's rule on the power.

LCSphysicist
That's what I got. The way I usually deal with limits of this form is to express it as a function of the exponential function and the natural logarithm, then use l'Hopital's rule on the power.
Yes but, what is wrong in the second case?

Eclair_de_XII
I think that the problem is that ##a## is not bounded, and it would not be the same as the limit for Euler's number if you tried to put it that way. For example, if you let ##a=\frac{1}{n^2}##, then as ##a\rightarrow \infty##, ##n^2\rightarrow 0^+##. Then your second limit becomes:

##\lim_{a \rightarrow \infty}(1+a)^{\frac{1}{a}}=\lim_{n^2\rightarrow 0}(1+\frac{1}{n^2})^{n^2}\neq e##

Maybe if you switched around the negative signs in the equation in the Relevant Equations part of your post, it would be some function of Euler's number.

• LCSphysicist
Homework Helper
Gold Member
Euler's number is ##\lim (1+\frac{1}{n})^n ## as ##n\to \infty## or equivalently ##\lim (1+n)^{\frac{1}{n}}## as ##n\to 0##.

BUT you have ##\lim (1+n)^{\frac{1}{n}}## as ##n\to \infty##

• • SammyS, FactChecker and LCSphysicist
LCSphysicist
Euler's number is ##\lim (1+\frac{1}{n})^n ## as ##n\to \infty## or equivalently ##\lim (1+n)^{\frac{1}{n}}## as ##n\to 0##.

BUT you have ##\lim (1+n)^{\frac{1}{n}}## as ##n\to \infty##
Oh so i was wrong to think that the third therm was too equivalent.
I really messed me up, the terms between the parentheses need to tend to one, not to infinity... thank you both guys

• Delta2