A continuous function is one that is predictable, has no breaks/jumps/holes, and is able to be drawn without lifting your pencil from the paper. In other words, you can predict where a continuous function is 'going' on the graph and there aren't any discontinuities (not to be confused with changes in equation) in the graph.
This is an example of a continuous function graph. As you can see, although it does change direction, the graph is still predictable after the change in equation, there aren't any discontinuities, and it is still able to be drawn without lifting pencil from paper.
A discontinuity is what makes a graph not continuous; it is a break, jump, or hole in the graph. There are four different types of discontinuities, separated into two families. The first family is the removable discontinuity family and it contains only point discontinuities, also known as holes.
This is a hole.
The second family is the non-removable discontinuity family, and it contains jump discontinuities, oscillating discontinuities, and infinite discontinuities.
The reason there are two families has to do with the limits (or lack thereof) of the discontinuities, which I will get to in the next question.
2. What is a limit? When does a limit exist? When does a limit does not exist? What is the difference between a limit and a value?
A limit is the intended height of the function- that is, it's where the limit wants to go. Sometimes it makes it there, and that's called a value. Continuous functions always have their limits and values equal each other. With removable discontinuities, it still reaches where it intends to go, but there isn't a value there; there's a hole. The limit of a point discontinuity doesn't match up with its value (which by the way is undefined), but there is still a limit. For non-removable discontinuities, the two sides of the function don't meet up at all and there is no limit. In this case, the limit Does Not Exist (or DNE for short) because the left side and right side of the function either approach different places (jump), approach so many places it becomes too wiggly to tell (oscillating), or has unbounded behavior caused by the presence of a vertical asymptote (infinite.)
3. How do we evaluate limits numerically, graphically, and algebraically?
To evaluate a limit numerically, we use a table like the one below.
We put what the limit approaches in the top center, go one tenth/hundredth/thousandth before and after it, and plug those numbers into the expression to find f(x) for each x value.
To evaluate a limit graphically, we use (big surprise) a graph. We look to see if the left and right sides of the function meet up. If they do, there is a limit (even if there's a hole.) If they don't, there is no limit and the reason depends on the discontinuity.
To evaluate a limit algebraically, we have three choices: direct substitution, dividing out/factoring, and rationalizing/conjugate.
The first method is direct substitution, which is exactly what it sounds like: taking whatever x approaches and plugging it into the expression. You should always use this method first because if you get a number over a number, a number over zero (aka undefined), or zero over a number (aka just plain zero), you're done! But if you get the dreaded zero over zero, that means there's a hole in the graph and you still need to do more work to figure out if the limit exists or not.
The second method is the dividing out/factoring method, which is also what it sounds like: looking to see if the numerator or the denominator has something to factor out and cancel. This removes the zero (and by extension, the hole, which is why point discontinuities are in the removable family.) After you've factored and canceled, plug in a la direct substitution and solve.
If direct substitution produces an indeterminate answer and factoring doesn't work, your last option is rationalizing/conjugate method. Multiply both the numerator and the denominator by the conjugate of the binomial term. The important thing to remember is NOT to factor the other term with the conjugate as the point of this is to get something to cancel, and it's easier to see if something is able to be canceled if you don't factor the other term with the conjugate. After you've FOILed the conjugates and canceled, use direct substitution and solve like you would normally.
Resources:
