Why are the asymptotes represented by ? Well, if we re-arrange the equation of the curve, becomes .
Now think about what happens if we calculate for large values of . Suppose, for instance, we take a million: that's . Then , a million million. To calculate , we'd then divide this by (which is presumably something fairly small. If , say, then .). Then from this still very large number, we subtract . As if that's going to make a lot of difference! Taking the square root and multiplying by completes the job.
Since subtracting the makes very little difference, for large values of , the values of will be very close to or . In fact, the bigger we make , the closer the answer will be. That's what makes the asymptotes.
Does that help to make it clearer?