# A limit question

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## Homework Statement

I was doing a problem and got to the following limit:
$$lim_{x \rightarrow 0^{+}} x^{\frac{1}{x} - 1}$$ I calculated it and got 0 but when I calculated it here:
It said that it wans't defined. Am I right?
Thanks.

## The Attempt at a Solution

$$lim_{x \rightarrow 0^{+}} x^{\frac{1}{x} - 1}=\lim_{x\rightarrow 0^{+}}e^{ln(x)^{\frac{1}{x}-1}}=\lim_{x\rightarrow\ 0^{+}} e^{\frac{1-x}{x}ln(x)}=e^{\lim_{x\rightarrow\ 0^{+}}\frac{1-x}{x} \lim_{x\rightarrow\ 0^{+}}ln(x)}=e^{\infty*(-\infty)}=e^{-\infty}=0$$