Infinite root out of x

Question

I and my friend were sitting in the maths lesson and we suddenly came up with what infinite root out of x is equal to. And we came up with this this equation:

$$ \sqrt[\infty]{x} = 1+ (x-1)×10^{-\infty} $$

I wonder if this makes any sense. (Im 10th former, and sorry if this is an out of topic question)

Answer 1

Let $x>0$. Write it as

$$ \displaystyle\lim_{n\to \infty}\sqrt[n]{x}=\lim_{n\to \infty} x^{1/n} $$ You see that the power $1/n$ goes to $0$ when $n$ increases so $x^{1/n}$ goes to $x^0$ which is $1$.

Answer 2

$\sqrt[\infty]{x}$ can be interpreted as

$\lim\limits_{a\rightarrow\infty}\sqrt[a]{x}$

and $1+(x-1)\cdot 10^{-\infty}$ can be interpreted as

$\lim\limits_{a\rightarrow\infty}(1+\large\frac{x-1}{10^a})$

In this sense the given equation is true since

$\lim\limits_{a\rightarrow\infty}\sqrt[a]{x}= \lim\limits_{a\rightarrow\infty}(1+\large\frac{x-1}{10^a})=1$