If `lim_(xto oo){(x^2+1)/(x+1)-(ax+b)}to oo`, then

Options

A. `a in (1,oo)`
B. `a ne 1 , b in R`
C. `a in (-oo,1)`
D. none of these

Explanation

To solve the limit problem given, we need to analyze the expression:

\[ \lim_{x \to \infty} \left( \frac{x^2 + 1}{x + 1} - (ax + b) \right) \]

### Step 1: Combine the terms

First, we need to find a common denominator for the expression. The common denominator will be \(x + 1\):

\[ \lim_{x \to \infty} \left( \frac{x^2 + 1}{x + 1} - (ax + b) \right) = \lim_{x \to \infty} \left( \frac{x^2 + 1 - (ax + b)(x + 1)}{x + 1} \right) \]

### Step 2: Expand the expression in the numerator

Next, we expand the term \((ax + b)(x + 1)\):

\[ (ax + b)(x + 1) = ax^2 + ax + bx + b = ax^2 + (a + b)x + b \]

Now, substitute this back into the limit:

\[ \lim_{x \to \infty} \left( \frac{x^2 + 1 - (ax^2 + (a + b)x + b)}{x + 1} \right) \]

### Step 3: Simplify the numerator

Now, simplify the numerator:

\[ x^2 + 1 - ax^2 - (a + b)x - b = (1 - a)x^2 - (a + b)x + (1 - b) \]

Thus, we have:

\[ \lim_{x \to \infty} \frac{(1 - a)x^2 - (a + b)x + (1 - b)}{x + 1} \]

### Step 4: Analyze the limit

As \(x\) approaches infinity, the highest degree term in the numerator will dominate the behavior of the limit. The degree of the numerator is 2 (due to the \(x^2\) term), and the degree of the denominator is 1.

For the limit to approach infinity, the coefficient of the highest degree term in the numerator must be positive:

\[ 1 - a > 0 \implies a < 1 \]

### Conclusion

Thus, for the limit to tend to infinity, the condition we derived is:

\[ a < 1 \]