Vertical assymptotes are the lines of the type x = 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 != 0
x != 7
So 7 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 - 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 = b that are basically the value the function start to take whenever x becomes bigger and bigger. Slant assymptotes (of the type y = 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 b, that is infinitely big, plug it in and see what happens.
y = b/(b-7)
Since b is an infinitely large number, we can say that b - 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/b or y ~= 1
So the horizontal assymptote is y = 1, that is, as x grows bigger the function gets closer and closer to 1.
We can doublecheck it by looking at the graph:
graph{x/(x-7) [-14, 14, -28, 28]}
