Limits+Portfolio

= **Li ** **mits Portfolio ** =

=
**1. What is the proper notation and how is it read correctly? Describe what a limit is and how to determine it with a graph, table and algebraically. Create examples to illustrate each method.** ====== A limit is the f(x) value you get when you plug in an x value. You can also find a limit when you plug in a number directly above and below the undefined point in the table of your graph. A limit can be defined in many different ways. Below are 3 different ways to find a limit: __Table:__ A limit is also the closest point of the whole on the graph without being undefined. __Graph:__ You can plug in any X value into the given equation and you get a corresponding f(x) value.  __Algebraically:__ When you plug in any number into the equation and get an answer. 
 * x || f(x) ||
 * 2 || 4.8 ||
 * 0 || undefined ||
 * -2 || 5.3 ||

=
**2. What behavior do you look at to determine what a limit is for a given function (even if it does not exist)? When does a limit exist? When does it not exist? What is a one-sided limit? How does one-sided limits affect whether a function has a limit? Create graphs and/or table that support your answers to these questions.** ====== To determine if a graph's given limit, you must look to see if there is a hole in the graph or if the graph has an asymptote. It also helps to determine if the graph is continuous or not. A limit exists when both the right and left hand sides of the graph arrive at the same point. A limit does not exist when both the right and left hand sides of the graph do not arrive at the same point. A one-sided limit is when you come from both sides, either the positive or negative. One-sided limits affect whether a function has a limit because if they arrive at the same point, there is a two-sided limit.

**3.How do functions that have holes or asymptotes affect limits? When does a limit go to infinity? What happens to certain limits as you make //x// infinitely large/small? Create graphs/tables that support your conclusions to these questions.** Functions that have holes or asymptotes affect limits because although a limit may exist, it isn't possible for you to plug in a number into the equation and solve. To find a limit for these types of problems, you must look at the graph's table. When a limit goes to infinity, not only does the graph go on and on, but it also moves towards zero (0). When the x value gets inifinitely larger for a fraction, you head towards zero (0). On the other hand, as the x value gets infinitely smaller for a fraction, you head toawrds infinity. Here is an example of a table that will help you to indentify a graph's limit. Notice that the two values that sandwich the undefined point are close to one specific number. That number, which is 7, is the limit. limit= 7
 * x || f(x) ||
 * 1 || 6.8 ||
 * 0 || undefined ||
 * -1 || 7.3 ||

<span style="font-family: Arial,Helvetica,sans-serif;">**4. How does a limit affect the conditions for continuity? Give examples of functions that are/are not continuous and describe the particular aspect of the conditions for continuity that are affected in each.** <span style="font-family: Arial,Helvetica,sans-serif;">A limit affects the conditions for continuity by equaling the the value of (c). A function will be continuous at a point if and only if it is continuous from both sides at that point. A non continuous is a point where a function is not continuous. A function is continuous on (a,b) if it is continuous at every point of the interval

<span style="font-family: Arial,Helvetica,sans-serif;"> If //f//(//x//) is defined on an open interval containing //c//, then //f//(//x//) is said to be **continuous at //c// ** if and only if <span style="font-family: Arial,Helvetica,sans-serif;">.

<span style="font-family: Arial,Helvetica,sans-serif; line-height: 0px; overflow: hidden;"> <span style="font-family: Arial,Helvetica,sans-serif;">This graph shows how the function does not exist, therefore the graph has no continuity because there is no way to redefine // k // at one point so that it will be continuous at 0.

<span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif; margin: 0in 0in 0pt 0.5in;"><span style="font-family: arial,helvetica,sans-serif;"> <span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif; margin: 0in 0in 0pt 0.5in;">