Vertical assymptotes are the lines of the type x = ax=a that the function can never take because doing so will create a math error.
In this function the only possible math error is a division by zero, so we have that:
x - 7 != 0x−7≠0
x != 7x≠7
So 77 is a vertical assymptote. That means that as the function gets closer and closer to 7, the values of the function as whole will become bigger in magnitude (because x - 7x−7 gets closer and closer to 0) but the function will never actually evaluate at that point.
Horizontal assymptotes are the lines of the type y = by=b that are basically the value the function start to take whenever xx becomes bigger and bigger. Slant assymptotes (of the type y = ax+by=ax+b) are very similar but the function isn't tending to a constant value. We can discover them the same way:
Start plugging bigger and bigger values until you see a pattern, or, use a made-up number, like, let's say bb, that is infinitely big, plug it in and see what happens.
y = b/(b-7)y=bb−7
Since bb is an infinitely large number, we can say that b - 7 ~= bb−7≅b, after all think of it like if b was a number like a billion or a trillion. A trillion minus 7 is still pretty much a trillion for practical purposes. So we have
y ~= b/by≅bb or y ~= 1y≅1
So the horizontal assymptote is y = 1y=1, that is, as xx grows bigger the function gets closer and closer to 11.
We can doublecheck it by looking at the graph:
graph{x/(x-7) [-14, 14, -28, 28]}
