H. Asymptotes for radicals

[WStephens]

Hi everyone,

I'm learning about Asymptotes of radicals for limits at infinity. I'm having a bit of trouble with the algebra of the work.

I can make sense of going b to c and then =1. (As x ->inf, 1/x^2 approaches 0 and thus the fraction = 1/1).

However I'm having difficulty getting from a to b. I know it's something simple but I'm blind to it right now.

Thanks

 

[Dr.Peterson]

I'll add in a step or two:

[MATH]\frac{x}{\sqrt{x^2-1}} = \frac{x}{\sqrt{x^2\left(1-\frac{1}{x^2}\right)}} = \frac{x}{\sqrt{x^2}\sqrt{1-\frac{1}{x^2}}} = \frac{x}{x\sqrt{1-\frac{1}{x^2}}} = \frac{1}{\sqrt{1-\frac{1}{x^2}}}[/MATH]
We factored out [MATH]x^2[/MATH] from the radicand, then pulled that outside the radical. Of course, that last step assumes that x is positive.

 

[WStephens]

Excellent, I now understand

Thank you very much.