How I find the range of the following, without graphing? $\ \Huge \mathsf{\frac{x}{3x-1}} ? $

Question

anonymous · Answer

well the range is everywhere that y is defined.. Since this is a fraction the range is defined where the denominator is not 0.. so set 3x-1= 0 and solve for x

anonymous · Answer

So...

anonymous · Answer

That gives me the $\ \mathsf{Domain}$ though, which is $\ \Huge \mathsf{(-infin,\frac{1}{3})U(\frac{1}{3}, infin)}...$

anonymous · Answer

you need to consider extreme values. 
consider this:
$$\frac{x}{3x -1} = \frac{1}{3 - \frac{1}{x}}$$

what happens when x has a very large magnitude?

anonymous · Answer

The 1/x gets closer to infity, so it's ~1/3

anonymous · Answer

does it ever  equal 1/3 ?