5.7. Without the use of a calculator, evaluate $$ \lim _{x \rightarrow 1} \arcsin \left(\frac{1-\sqrt{x}}{1-x}\right) $$.

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1. Begin with the limit:
   $$
   \lim_{x \to 1} \arcsin \left(\frac{1-\sqrt{x}}{1-x}\right)
   $$

2. Focus on simplifying the expression inside the $$\arcsin$$:
   $$
   \frac{1-\sqrt{x}}{1-x}
   $$
   Notice that as $$x \to 1$$, both the numerator and denominator tend to 0. To simplify, multiply the numerator and denominator by $$1+\sqrt{x}$$:
   $$
   \frac{1-\sqrt{x}}{1-x} = \frac{1-\sqrt{x}}{1-x} \cdot \frac{1+\sqrt{x}}{1+\sqrt{x}} = \frac{(1-\sqrt{x})(1+\sqrt{x})}{(1-x)(1+\sqrt{x})}
   $$

3. Recognize that the numerator is a difference of squares:
   $$
   (1-\sqrt{x})(1+\sqrt{x}) = 1 - x
   $$
   So the expression simplifies to:
   $$
   \frac{1-x}{(1-x)(1+\sqrt{x})} = \frac{1}{1+\sqrt{x}}
   $$
   This cancellation is valid for $$x \neq 1$$.

4. Now substitute $$x \to 1$$ into the simplified expression:
   $$
   \lim_{x \to 1} \frac{1}{1+\sqrt{x}} = \frac{1}{1 + \sqrt{1}} = \frac{1}{1+1} = \frac{1}{2}
   $$

5. Therefore, the original limit becomes:
   $$
   \arcsin\left(\frac{1}{2}\right)
   $$

6. Recall that:
   $$
   \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}
   $$
   Hence:
   $$
   \arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}
   $$

7. The final answer is:
   $$
   \boxed{\frac{\pi}{6}}
   $$
The limit evaluates to $$ \frac{\pi}{6} $$.

5.7. Without the use of a calculator, evaluate .

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