A simple question on limits

Question

Does the following equation hold:

$$ \lim_{x\to\infty} x^x = \left(\lim_{x\to\infty} x\right)^{\!x}?$$

I know that

$$ \lim_{x\to\infty} x^n = \left(\lim_{x\to\infty} x\right)^{\!n}$$

holds if $n$ is different from $x$, but I am not sure if it holds where $n=x$.

Answer 1

limit is similar to function, when you take limit, you let $x$ go to a specific point, or infinity, and you will get a exactly ONE answer if limit exist.( limit is unique if the space is Hausdorff, $R^n$ is Hausdorff).

I will slightly modify your question to :

$\lim_{x\to\infty} (\frac{1}{x}) ^x = (\lim_{x\to\infty} (\frac{1}{x}))^{x}$?

if we calculate the limit, we have $0=0^x$.

for the right hand side of the equality, without any information on $x$, we cannot compute anything.

Answer 2

A notation like

$$\left(\lim_{x\to\infty}x\right)^x$$ is twice meanigless because

• $x$ is used both as an independent variable (the exponent) and a dummy variable (argument of the limit), so which is which is ambiguous;

• the limit diverges so that raising it to a power is not defined.

Answer 3

No, they are not the same.

We have two expressions here.$$\lim_{x\to\infty} x^x$$

and $$ \left(\lim_{x\to\infty} x\right)^{\!x}$$

The first expression makes perfect sense and the answer is $$\lim_{x\to\infty} x^x = \infty $$

The second expression is ambiguous.

You have $$\lim_{x\to\infty} x =\infty$$ and you want to raise it to power of $x$.

We have to know what is $x$ here to make an inference about $(\infty )^x.$

Of course we get a different answer for positive $x$ and for negative $x$.

Thus $$ \left(\lim_{x\to\infty} x\right)^{\!x}$$ depends on $x$.