Solve (i) `lim_(xto1^(-))sin (sin^(-1)x)` (ii) `lim_(xto pi//2)sin^(-1)(sinx)`

Answer

Let's solve the given limits step by step.

### (i) Solve `lim_(x -> 1^(-)) sin(sin^(-1)x)`

1. **Understanding the Limit**: We need to find the limit of `sin(sin^(-1)x)` as `x` approaches `1` from the left (denoted as `1^(-)`).

2. **Substituting the Limit**: As `x` approaches `1` from the left, we can express it as:
\[ \lim_{x \to 1^-} \sin(\sin^{-1} x) \]
Since `sin^{-1} x` is the inverse sine function, we know that `sin^{-1}(1) = \frac{\pi}{2}`.

3. **Applying the Limit**: Now, substituting `x = 1` into the expression:
\[ \sin(\sin^{-1}(1)) = \sin\left(\frac{\pi}{2}\right) \]

4. **Calculating the Sine**: We know that:
\[ \sin\left(\frac{\pi}{2}\right) = 1 \]

5. **Final Result**: Therefore, the limit is:
\[ \lim_{x \to 1^-} \sin(\sin^{-1} x) = 1 \]