Unit U Big Question #1

1. What is continuity? What is discontinuity?

A graph that is continuous means we can draw the entire thing without lifting our pencil from the paper. It has no holes, no breaks, no jumps in the graph. The continuous graph is predictable and makes a good bridge because a person "driving" over it would not fall through or fall off.

On the other hand....a discontinuous graph is the exact opposite. We have four types of discontinuities divided into two families.

• Removable Discontinuities
• Wherever there is a HOLE in the graph, it is called a point discontinuity.
  
• Non-Removable Discontinuities
• Jump discontinuities includes a break in the graph. 
• Oscillating behavior occurs when the graph is "wiggly" and unpredictable.
• Infinite discontinuity occurs whenever there is a vertical asymptote and is also known as unbounded behavior.
  

(Images from education.com, wyzant.com, and en.wikipedia.org)

Unit U Big Question #2

2. What is a limit? 

A limit is the intended height of a function at a certain value. Basically what this means is it is the y-value at which a function is approaching on the x-axis but may or may not reach it but it is intended to reach it.

 When does a limit exist? When does a limit not exist?

A limit exists as long at the same height (y-value) is reached from both the left and the right. In other words, the limit exists if both the right hand limit and the left hand limit are the same. If the graph does not break at a certain x-value, then the limit exists. For instance, at a hole, the limit exists because the intended height was still reached, even if the "diner" (in the car analogy) is not there.

 

   

 

(Image from curvebank.calstatela.edu)

What is the difference between a limit and a value? 

 

(Image from Unit U SSS packet)  

 The limit is the intended height and the value is the real or actual height. As you can see in the picture above, the limit and value are two different things. We can have the limit equal the value on a continuous graph because the limit is always reached and there are not breaks OR holes in the graph.

Unit U Big Question #3

3. How do we evaluate limits numerically, graphically, and algebraically?

NUMERICALLY

When we evaluate limits numerically, it is on a table. We are finding the intended height of the function, which is the y-value as the graph, and it is referred to as the LIMIT. We use a table similar to the one in the picture above. As x is approaching a certain value, in this case 3, we write numbers approaching the limit from the left and from the right. This way, we ALWAYS find what the limit is intending to reach even if the graph never actually reaches that point. (Image from Unit U SSS Packet)

GRAPHICALLY

(Images from www.education.com)

When we evaluate limits graphically, we are looking at a graph and using our fingers to approach the limit from the left and right to see if they meet.

ALGEBRAICALLY

There are three ways of evaluating limits algebraically: direct substitution method, dividing out/factoring method, and rationalizing/conjugate method.   

1.  Direct substitution is the method used first. We take the limit and plug in that number into all the x's in the equation to find an answer. We can get a numerical answer which is what we hope for and we are finished. We can get 0/# and this means our limit is 0 and we are finished. We can get #/0 which means the limit is undefined and DNE because of unbounded behavior. And lastly, we can get 0/0 which means we are not finished. 0/0 is indeterminate and we need to use one of the other methods to evaluate it.

2. Dividing out/factoring method is used after we get indeterminate form (0/0).  We factor the numerator and denominator and cancel any common terms on top and bottom. When we do this, we are removing the zero in the denominator. Then, we use direct substitution with the simplified expression. This is similar to finding a hole in a past unit.

3. Rationalizing/conjugate method is used when the fraction has a radical in it. We use the conjugate of either the numerator or the denominator depending on where the radical is. We foil the conjugates and leave the non-conjugate factored. Then, we use direct substitution.