Answer to Question #223292 in Calculus for Ramon

Question

Evaluate the limit as x turns to -∞

Lim [3x + √(9x²-Ix-2I)]

Answer 1

$$\lim\limits_{x\to-\infin}(3x+\sqrt{9x^2-|x-2|})$$

$$=\lim\limits_{x\to-\infin}(3x+\sqrt{9x^2-|x-2|})\cdot\dfrac{3x-\sqrt{9x^2-|x-2|}}{3x-\sqrt{9x^2-|x-2|}}$$

$$=\lim\limits_{x\to-\infin}\dfrac{9x^2-9x^2+|x-2|}{3x-\sqrt{9x^2-|x-2|}}$$

$$=\lim\limits_{x\to-\infin}\dfrac{\dfrac{|x-2|}{|x|}}{\dfrac{3x}{|x|}-\sqrt{\dfrac{9x^2}{x^2}-\dfrac{|x-2|}{x^2}}}$$

$$=\dfrac{1}{-3-\sqrt{9-0}}=-\dfrac{1}{6}$$