If limx→∞ x2+x+1x+1−ax−b=4 , then - Infinity Learn by Sri Chaitanya

A
$a = 1, b = 4$
B
$a = 1, b = - 4$
C
$a = 2, b = - 3$
D
$a = 2, b = 3$

Solution:

We have,

$\begin{array}{l} lim_{x \to \infty} (\frac{x^{2} + x + 1}{x + 1} - a x - b) = 4 \\ \Rightarrow lim_{x \to \infty} \frac{x^{2} (1 - a) + x (1 - a - b) + 1 - b}{x + 1} = 4 \end{array}$

lf $f (x)$ and $g (x)$ are polynomials such that $lim_{x \to \infty} \frac{f (x)}{g (x)}$ is a finite

non-zero number, then $f (x)$ and $g (x)$ must be of the same

degree. Therefore $x^{2} (1 - a) + x (1 - a - b) + 1 - b$ and $x + 1$

must be of the same degree. This is possible only when $a = 1$ . ln that case

$\begin{array}{l} lim_{x \to \infty} \frac{x^{2} (1 - a) + x (1 - a - b) + 1 - b}{x + 1} = 4 \\ \Rightarrow lim_{x \to \infty} \frac{- x b + 1 - b}{x + 1} = 4 \\ \Rightarrow - b = 4 \Rightarrow b = - 4 \end{array}$

Hence, $a = 1$ and $b = - 4$

If $lim_{x \to \infty} (\frac{x^{2} + x + 1}{x + 1} - a x - b) = 4$ , then

Solution:

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