Find the following limits: `(i) lim_(xto0) (1-x)^((1)/(x))" "(ii) lim_(xto1) (1+log_(e)x)^((1)/(log_(e)x))`
`(iii)lim_(xto0) (1+sinx)^((1)/(x))`

Explanation

Let's solve the given limits step by step.

### (i) Find \( \lim_{x \to 0} (1 - x)^{\frac{1}{x}} \)

1. **Identify the form**: As \( x \to 0 \), \( 1 - x \to 1 \) and \( \frac{1}{x} \to \infty \). This limit is of the form \( 1^{\infty} \).

2. **Rewrite the limit**: We can express the limit as:
\[
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