First slide

Introduction to limits

Question

$lim_{x \to - \infty} (3 x + \sqrt{9 x^{2} - x})$ equals

Moderate

A

1 / 3
B

1 / 6
C

-1 / 6
D

-1 / 3

Solution

We have,
$\begin{array}{l} lim_{x \to - \infty} (3 x + \sqrt{9 x^{2} - x}) \\ = lim_{y \to \infty} (- 3 y + \sqrt{9 y^{2} + y}), where y = - x \end{array}$
$= lim_{y \to \infty} \frac{- 9 y^{2} + 9 y^{2} + y}{(3 y + \sqrt{9 y^{2} + y)}} = lim_{y \to \infty} \frac{y}{3 y + \sqrt{9 y^{2} + y}} = \frac{1}{3 + 3} = \frac{1}{6}$ .

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