**Continuity Vs. Discontinuity**

1. Continuity is when the function has no breaks, holes, or jumps. They can be written/drawn without lifting your pencil off the paper.

<-- (they look like this)

Discontinuities can't be written without lifting your pencil. Discontinuities are separated into two different families: removable and non- removable discontinuities. The only discontinuity in removable Discontinuities is the point discontinuity.

Point Discontinuity- Known as a hole. The limit still exists but it isn't the same as the actual value since the limit is the intended height of the function and the value is the actual height.

Non Removable Discontinuities:

Jump, Oscillating, Infinite discontinuity (unbounded behavior)

2.**Limits and Values**

Limits are the intended height of the function and the value is the actual height of the function. When the value and the limit are the same that means that the graph is continuous.

The limit does exist in Removable Discontinuities because even though the value might be undefined or an actual number, (depending if its just a whole or the actual point is somewhere else on the graph) the graph still has an intended height.

The limit doesn't exist at Non-removable Discontinuities because it doesn't approach a single value at all.

**3. V.A.N.G**

V- verbally which is more like actually speaking it out so it isn't too hard.

A- algebraically. There are three different ways to evaluate limits.

-Direct substitution- This concept is easiest in all honesty. Nothing really hard about grabbing your calculator and just plugging in what approaches f(x) into the calculator. You can either get 0/(any number) or (any number)/0 (undefined.) When it comes out being 0/0, this is when you have a problem. This is called indeterminate form and you have to use either dividing out/ factoring out or the rationalizing/ conjugate method.

-Dividing out/ Factoring out Method- In this method, you have to factor both the numerator and denominator and cancel out some common terms to remove the zero in the denominator. Then use **direct substitution with the simplified expression.** This is used for removable discontinuities to actually remove the discontinuity.

-Rationalizing/Conjugate Method- The conjugate is where you change the sign in the middle of two terms (3x+1 changes to 3x-1). Sometimes we use the Conjugate of the denominator while other times we use the conjugate of the numerator. (It depends on where the radical is.) Only foil the conjugate with the original part not the conjugate with the non conjugate. (we hope something will cancel out.)

N-Numerically.

With this concept we just use a graph to find the limit. If the limit can be reached then the limit would be in the middle. If it can't be reached that means it doesn't exist Because it never reaches a certain value. the Limit from the left and the right just get really really really really really really really close.

G-Graphically. You can just plug it into the calculator and hit trace with a certain value. If it is error then the limit can't be reached. But if it does give you a number then the limit can be reached. Another way is to put your left finger on the left side of the graph and your right finger on the right side of the graph. If your fingers meet then the limit exists but if not then the limit does not exist due to different L/R (left / right)

These pictures came from the SSS packets from the wonderful Mrs. Kirch. :) They can be found __here.__