First slide

Introduction to limits

Question

The value of the constants $α$ and $β$ such that $lim_{x \to \infty} (\frac{x^{2} + 1}{x + 1} - αx - β) = 0$ are , respectively

Moderate

A

(1,1)
B

(1,-1)
C

(-1,1)
D

(0,1)

Solution

Given, $lim_{x \to \infty} (\frac{x^{2} + 1}{x + 1} - αx - β) = 0$
$\begin{array}{l} \Rightarrow lim_{x \to \infty} (\frac{x^{2} + 1 - α (x^{2} + x) - β (x + 1)}{x + 1}) = 0 \\ \Rightarrow lim_{x \to \infty} (\frac{2 x - α (2 x + 1) - β (1)}{1}) = 0 \end{array}$
[using L' Hospital's rule]
lf this limit is zero, then the function
$\begin{aligned} 2 x - α (2 x + 1) - β = 0 \\ \Rightarrow x (2 - 2 α) - (α + β) = 0 \end{aligned}$
Equating the coefficient of x and constant terms, we get
$\begin{array}{l} 2 - 2 α = 0 \\ and α + β = 0 \\ \Rightarrow α = 1, β = - 1 \end{array}$

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