`lim_(x to 0) (sqrt(1 + x + x^(2)) - sqrt(x + 1))/(2X^(2))` is equal to

Options

A. `(1)/(6)`
B. `(1)/(4)`
C. `(3)/(2)`
D. `(9)/(2)`

Answer

To solve the limit

\[ \lim_{x \to 0} \frac{\sqrt{1 + x + x^2} - \sqrt{1 + x}}{2x^2}, \]

we will follow these steps:

### Step 1: Rewrite the Expression
We start by rewriting the limit without the limit notation for clarity:

\[ \frac{\sqrt{1 + x + x^2} - \sqrt{1 + x}}{2x^2}. \]

### Step 2: Multiply by the Conjugate
To simplify the expression, we multiply the numerator and denominator by the conjugate of the numerator:

\[ \frac{(\sqrt{1 + x + x^2} - \sqrt{1 + x})(\sqrt{1 + x + x^2} + \sqrt{1 + x})}{2x^2(\sqrt{1 + x + x^2} + \sqrt{1 + x})}. \]

This gives us:

\[ \frac{(1 + x + x^2) - (1 + x)}{2x^2(\sqrt{1 + x + x^2} + \sqrt{1 + x})}. \]

### Step 3: Simplify the Numerator
Now, simplify the numerator:

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

So, we have:

\[ \frac{x^2}{2x^2(\sqrt{1 + x + x^2} + \sqrt{1 + x})}. \]

### Step 4: Cancel \(x^2\)
Now we can cancel \(x^2\) in the numerator and denominator:

\[ \frac{1}{2(\sqrt{1 + x + x^2} + \sqrt{1 + x})}. \]

### Step 5: Evaluate the Limit
Now we can evaluate the limit as \(x\) approaches 0:

\[ \lim_{x \to 0} \frac{1}{2(\sqrt{1 + 0 + 0^2} + \sqrt{1 + 0})} = \frac{1}{2(\sqrt{1} + \sqrt{1})} = \frac{1}{2(1 + 1)} = \frac{1}{4}. \]

### Final Answer
Thus, the limit is:

\[ \frac{1}{4}. \]

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