[tex]\bf \lim\limits_{x\to -\infty}~\cfrac{\sqrt{9x^2+2}}{2x-9}\implies \cfrac{\lim\limits_{x\to -\infty}~\sqrt{9x^2+2}}{\lim\limits_{x\to -\infty}~2x-9}[/tex]
now, for any "x" value, the numerator will give you a positive root, say x = -10, then 9(-10)^2 +2 will be 902, square root of that is about 30.
whilst for the denominator, that'd be 2(-100) -9, or -209.
as you can see, the denominator for any "x" value, has larger jumps then the numerator, and because of that the rational itself, will always have a larger and larger and negative denominator.
so it may go from say -1/100 to -1/1000000 and so on as "x" progresses, going ever closer and closer to 0.
