If `L=lim_(xtooo) (x+1-sqrt(ax^(2)+x+3))` exists infinetely then
The value of a is

Explanation

To solve the limit problem given, we need to find the value of \( a \) such that the limit

\[ L = \lim_{x \to \infty} \left( x + 1 - \sqrt{a x^2 + x + 3} \right) \]

exists. Here’s a step-by-step solution:

### Step 1: Write the limit expression
We start with the limit:

\[ L = \lim_{x \to \infty} \left( x + 1 - \sqrt{a x^2 + x + 3} \right) \]

### Step 2: Rationalize the expression
To simplify the expression, we can multiply and divide by the conjugate:

\[ L = \lim_{x \to \infty} \frac{(x + 1 - \sqrt{a x^2 + x + 3})(x + 1 + \sqrt{a x^2 + x + 3})}{x + 1 + \sqrt{a x^2 + x + 3}} \]

This gives us:

\[ L = \lim_{x \to \infty} \frac{(x + 1)^2 - (a x^2 + x + 3)}{x + 1 + \sqrt{a x^2 + x + 3}} \]

### Step 3: Expand the numerator
Now we expand the numerator:

\[ (x + 1)^2 = x^2 + 2x + 1 \]

So, we have:

\[ L = \lim_{x \to \infty} \frac{x^2 + 2x + 1 - (a x^2 + x + 3)}{x + 1 + \sqrt{a x^2 + x + 3}} \]

This simplifies to:

\[ L = \lim_{x \to \infty} \frac{(1 - a)x^2 + (2 - 1)x + (1 - 3)}{x + 1 + \sqrt{a x^2 + x + 3}} \]

### Step 4: Combine like terms
Combining the terms in the numerator gives:

\[ L = \lim_{x \to \infty} \frac{(1 - a)x^2 + (1)x - 2}{x + 1 + \sqrt{a x^2 + x + 3}} \]

### Step 5: Factor out \( x^2 \) from the square root
In the denominator, we can factor out \( x^2 \) from the square root:

\[ \sqrt{a x^2 + x + 3} = x\sqrt{a + \frac{1}{x} + \frac{3}{x^2}} \]

Thus, we rewrite the limit as:

\[ L = \lim_{x \to \infty} \frac{(1 - a)x^2 + (1)x - 2}{x + 1 + x\sqrt{a + \frac{1}{x} + \frac{3}{x^2}}} \]

### Step 6: Simplify the limit
Now, we can simplify the limit by dividing the numerator and denominator by \( x \):

\[ L = \lim_{x \to \infty} \frac{(1 - a)x + 1 - \frac{2}{x}}{1 + \frac{1}{x} + \sqrt{a + \frac{1}{x} + \frac{3}{x^2}}} \]

### Step 7: Analyze the limit as \( x \to \infty \)
For the limit to exist as \( x \to \infty \), the coefficient of \( x \) in the numerator must be zero:

\[ 1 - a = 0 \implies a = 1 \]

### Conclusion
Thus, the value of \( a \) is:

\[ \boxed{1} \]