How to prove the limit of n-th root of n is 1 by definition

In fact, definition is to do difference, proving that the absolute value of difference can be infinitely small. This way can not be said to work. In other words, it is impossible to simply use the definition to prove, so we have to use other ideas to go around. We can find ln (x times root sign of x) when x approaches + infinity, the limit is 0

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