The largest value of the non-negative integer a for which
`lim_(xto1) {(-ax+sin(x-1)+a)/(x+sin(x-1)-1)}^((1-x)/(1-sqrt(x)))=(1)/(4)` is ___________.

Explanation

To solve the limit problem, we need to find the largest non-negative integer \( a \) such that

\[ \lim_{x \to 1} \left( \frac{-ax + \sin(x-1) + a}{x + \sin(x-1) - 1} \right)^{\frac{1-x}{1-\sqrt{x}}} = \frac{1}{4}. \]

### Step 1: Simplifying the Limit Expression

First, we analyze the limit as \( x \) approaches 1. We can rewrite the limit expression:

\[ \lim_{x \to 1} \frac{-ax + \sin(x-1) + a}{x + \sin(x-1) - 1}. \]

As \( x \to 1 \), both the numerator and denominator approach 0, which suggests we can apply L'Hôpital's Rule or simplify the expression further.

### Step 2: Expanding \( \sin(x-1) \)

Using the Taylor series expansion for \( \sin(x-1) \) around \( x = 1 \):

\[ \sin(x-1) \approx (x-1) - \frac{(x-1)^3}{6} + O((x-1)^5). \]

Substituting this into our limit gives:

\[ \sin(x-1) \approx (x-1). \]

### Step 3: Substitute and Simplify

Now substituting \( \sin(x-1) \) into our limit expression:

- Numerator:
\[ -a(x-1) + (x-1) + a = (1-a)(x-1). \]

- Denominator:
\[ x + (x-1) - 1 = 2(x-1). \]

Thus, our limit expression simplifies to:

\[ \lim_{x \to 1} \frac{(1-a)(x-1)}{2(x-1)} = \frac{1-a}{2}. \]

### Step 4: Evaluating the Exponent

Next, we evaluate the exponent \( \frac{1-x}{1-\sqrt{x}} \) as \( x \to 1 \):

Using L'Hôpital's Rule:

\[ \lim_{x \to 1} \frac{1-x}{1-\sqrt{x}} = \lim_{x \to 1} \frac{-1}{-\frac{1}{2\sqrt{x}}} = 2. \]

### Step 5: Final Limit Expression

Now, we can rewrite our limit as:

\[ \left( \frac{1-a}{2} \right)^{2}. \]

Setting this equal to \( \frac{1}{4} \):

\[ \left( \frac{1-a}{2} \right)^{2} = \frac{1}{4}. \]

### Step 6: Solve for \( a \)

Taking the square root of both sides gives:

\[ \frac{1-a}{2} = \pm \frac{1}{2}. \]

This leads to two cases:

1. \( 1-a = 1 \) which gives \( a = 0 \).
2. \( 1-a = -1 \) which gives \( a = 2 \).

### Step 7: Conclusion

The non-negative integer values of \( a \) are \( 0 \) and \( 2 \). The largest value is:

\[ \boxed{2}. \]