`lim_(x to oo)(sqrt(x^2-x+1)-ax)=b` Find the value of `2(a+b)` is

Options

A. -3
B. -1
C. 3
D. 1

Answer

To solve the limit problem \( \lim_{x \to \infty} (\sqrt{x^2 - x + 1} - ax) = b \), we will follow these steps:

### Step 1: Rewrite the Limit Expression
We start with the limit expression:
\[ \lim_{x \to \infty} (\sqrt{x^2 - x + 1} - ax) = b \]

### Step 2: Rationalize the Expression
To simplify the limit, we can rationalize the expression by multiplying the numerator and denominator by the conjugate:
\[ \lim_{x \to \infty} \frac{(\sqrt{x^2 - x + 1} - ax)(\sqrt{x^2 - x + 1} + ax)}{\sqrt{x^2 - x + 1} + ax} \]
This gives us:
\[ \lim_{x \to \infty} \frac{x^2 - x + 1 - a^2x^2}{\sqrt{x^2 - x + 1} + ax} \]

### Step 3: Simplify the Numerator
Now, simplify the numerator:
\[ x^2 - x + 1 - a^2x^2 = (1 - a^2)x^2 - x + 1 \]

### Step 4: Analyze the Denominator
The denominator becomes:
\[ \sqrt{x^2 - x + 1} + ax \]
As \( x \to \infty \), we can approximate \( \sqrt{x^2 - x + 1} \) as \( x \) (since \( x^2 \) dominates):
\[ \sqrt{x^2 - x + 1} \sim x \]
Thus, the denominator simplifies to:
\[ x + ax = (1 + a)x \]

### Step 5: Combine and Simplify the Limit
Now we can rewrite the limit:
\[ \lim_{x \to \infty} \frac{(1 - a^2)x^2 - x + 1}{(1 + a)x} \]
Dividing every term in the numerator by \( x \):
\[ \lim_{x \to \infty} \frac{(1 - a^2)x - 1 + \frac{1}{x}}{1 + a} \]

### Step 6: Evaluate the Limit
As \( x \to \infty \), the terms \( -1 \) and \( \frac{1}{x} \) become negligible:
\[ \lim_{x \to \infty} \frac{(1 - a^2)x}{1 + a} = b \]
For this limit to exist and be finite, we need \( 1 - a^2 = 0 \), which implies:
\[ a^2 = 1 \quad \Rightarrow \quad a = 1 \text{ or } a = -1 \]

### Step 7: Determine the Value of \( b \)
If \( a = 1 \):
\[ b = \lim_{x \to \infty} \frac{0}{1 + 1} = 0 \]
If \( a = -1 \):
\[ b = \lim_{x \to \infty} \frac{(1 - 1)x}{1 - 1} \text{ (undefined)} \]
Thus, we take \( a = 1 \) and \( b = 0 \).

### Step 8: Calculate \( 2(a + b) \)
Now we find:
\[ 2(a + b) = 2(1 + 0) = 2 \]

### Final Answer
The value of \( 2(a + b) \) is:
\[ \boxed{2} \]