BQ #6 Unit U - Introduction to Limits

What is a continuity? What is a discontinuity?

Well for starters, a continuity is when the function is predictable (meaning you know where it's going), no jumps, breaks, or holes (essentially the non-removable discontinuities that we will discuss later on), and can be drawn without lifting your hand up. A discontinuity is divided into two families: removable discontinuities and non-removable discontinuities. The removable discontinuities consist of a point discontinuity. The point discontinuity is where a hole exist and it is also where the limit can also exist at the hole. As for the non-removable discontinuities, they consist of jump discontinuity which is a results from different left/right behaviors, oscillating behavior (which is a wiggly line where there is no real value), and the infinite discontinuity which has a vertical asymptote that results in unbounded behavior. A non-removable discontinuity is where the limit does not exist and the removable discontinuity is where the limit does exist.

What is a limit? When does a limit exist? When does a limit not exist? What is the difference between a limit and a value?

A limit is simply the intended height of the function and it exist at removable discontinuities that are also known as point discontinuities. The limit does not exist at non-removable discontinuities that are also known as jump, oscillating, and infinite discontinuities. Limits do not exist at non-removable discontinuities because of different left right behavior where the lines do not meet together at one point, or when there is no real value in the oscillating discontinuity where it does not approach a single value, or when the line reaches towards infinity because infinity is not defined as a real number. Limits do not also exist at the infinite discontinuity because of the vertical asymptotes that result in unbounded behavior.
The main difference between a limit and a value is that the limit is the intended height of the function while the graph is the actual height of the function.

How do we evaluate limits numerically, graphically, and algebraically, (VANG)?

Algebraically

By using direct substitution, factoring/dividing, or the conjugate method, we can solve for these limits algebraically. For direct substitution, it is simply plugging in whatever x approaches into the function. However, there are occasions where the result is an indeterminate form which is 0/0. To resolve this problem, we use the factoring/dividing method which removes the hole that causes the indeterminate form. Basically,  the factoring/dividing method is where we will factor out the function as best as we can and then divide/cancel out any like terms to get rid of the hole. As for the conjugate form, this is when there is a radical and we use the conjugate method to make our lives easier and get rid of the radical. However, it is important to remember that we should always use direct substitution method first.

Numerically

For finding limits numerically, we use a table. Usually, we will plug the function into the graphing calculator and hit trace to where the limit supposedly is. Since we can't actually reach the limit, we approximate as close as we can. For example the we can't reach the number 2 but we can get really really close to it so we write it as 1.9, 1.99, 1.999 or 2.1, 2.01, 2.001

Graphically

Graphically is probably the simplest method. To do this we basically just put our fingers on the graph to the left and right of where the limit is. From there we move our fingers closer to the limit. If our fingers touch then that means that we a limit> However, if our fingers do not touch for whatever reason, then that means we do not have a limit so we write DNE (does not exist).


Reference Unit U SSS - Online (Kirch's)