How are the locations of vertical asymptotes and holes different, and what role do limits play?
To talk about asymptotes and holes, you need pictures. These pictures are graphs of functions. The simplest function containing vertical and horizontal asymptotes is y=1/x, where x is the horizontal axis and y the vertical axis. The vertical asymptote is in fact the y axis, because the graph has no values that would quite plot onto the y axis, although the curve for 1/x gets very, very close. The reason is that the y axis represents x=0, and you can't evaluate 1/x when x=0. You' d have to extend the y axis to infinity both positively and negatively. You can see this if you put a small positive or negative value for x into the function. If x=1/100 or 0.01, y becomes 100. If x=-1/100 or -0.01, y becomes -100. If the magnitude of x decreases further y increases further. That's the vertical asymptote. It represents the inachievable.
What about the horizontal asymptote? The same graph has a horizontal asymptote. As x gets larger and larger in magnitude, positively or negatively, the fraction 1/x gets smaller and smaller. This means that the curve gets closer and closer to the x axis, but can never quite touch it. So, like the y axis, the axis extends to infinity at both ends. What does the graph look like? Take two pieces of thick wire that can be bent. The graph comes in two pieces. Bend each piece of wire into a right angle like an L. Because the wire is thick it won't bend into a sharp right angle but will form a curved angle. Bend the arms of the L out a bit more so that they diverge a little. Your two pieces of wire represent the curve(s) of the function. The two axes divide your paper into four squares. Put one wire into the top right square and the other into the bottom left and you get a picture of the graph, but make sure neither piece of wire actually touches either axis, because both axes are asymptotes. The horizontal axis represents the value of x needed to make y zero, the inverse function x=1/y. Hence the symmetry of the graph.
Any function in which an expression involving a variable is in the denominator of a fraction potentially generates a vertical asymptote if that expression can ever be zero. If the same expression can become very large for large magnitude values of the variable, potentially we would have horizontal asymptotes. I use the word "potentially" because there's also the possibility of holes under special circumstances. Asymptotes and holes are both no-go zones, but holes represent singularities and they're different from asymptotes. Take the function y=x/x. It's a very trivial example but it should illustrate what a hole is. Like 1/x, we can't evaluate when x=0. However, you might think you can just say y=1 for all values of x, since x divides into x, cancelling out the fraction. That's a horizontal line passing through the y axis at y=1. Yes, it is such a line except where x=0. We mustn't forget the original function x/x. So where the line crosses the y axis there's a hole, a very tiny hole with no dimensions, a singularity.
So a hole can occur when the numerator and denominator contain a common factor. If this common factor can be zero for a particular value of x, then a hole is inevitable. Effectively it's an example of the graphical result of dividing zero by zero. With functions we can't simply cancel common factors as we normally do in arithmetic.
Asymptotes and holes are examples of limits. Asymptotes can show where functions converge to a particular value without ever reaching it. Asymptotes can be slanted, they don't have to be horizontal or vertical, and they can be displaced from both axes. Graphs can aid in the solution of mathematical and physics problems and can reveal where limitations and limits exist for complicated and complex functions. Knowing where the limits are by inspection of functions also aids in drawing the graph. This helps in problems where the student may be asked to draw a graph to show the key features without plotting it formally. Read More: ...