AP+Calculus+Limit

[[image:http://upload.wikimedia.org/math/e/d/8/ed80e81395fb7b21643891fdd4190429.png caption=" \lim_{x \to c}f(x) = L "]]
This is read as: The limit of f of (x) as //x// approaches //c// is L. What is a limit? A limit is used to describe the value that a function approaches as the input approaches a value. For example if:  then //f //(1) is not defined, yet as //x // approaches 1, //f //(//x //) approaches 2:
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===** 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. **=== The behavior could vary. The graph is not continuous and could appear as an asymptote, removable discontinuity, and step discontinuity. A limit exist when the lines from both sides of x approaches the same point. If it does not approach the same point, the limit does not exist. A one sided limit occurs with the step discontinuity and asymptotes. or If both of these limits are equal to //L// then this can be referred to as **//the// limit of //f//(//x//) at //p//**. Conversely, if they are not both equal to //L// then //the// limit, as such, does not exist. Example: One sided limit

=== **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 in the graph are also known as removable discontinuity. Basically this means that the limit for the function is not continuous but it in facts has an APPROACHING limit. For example, in the graph below there is a hole in the graph when x = 2. Therefore, in this graph the limit is actually 2. It is a two sided removable discontinuity graph and the limit from both sides approaches 2. When a graph has an asymptote, it really means that the function will continue going but WiLL NOT pass the asymptote. It will go fairly close to it but will not reach it. For example, in the graph to the right, where the asymptote equals x=2 and y=4, no x value will actually equal 2, and no y value will actually equal 4. The values in this particular graph will come fairly close to those values, but NOT equal it.



A limit goes into infinity when the graph is continuous. As the limit becomes infinitely large, the limit will be defined as DNE. When a limit becomes infinitely small, the values will get smaller and smaller. For example. Consider a fraction 1/x. As x gets larger (goes to infinity), the fraction gets infinitely small.. Specifically, lim_(x->infinity) 1/x = 0 meaning the value of 1/x approaches zero but never quite gets there. I would call this an infinitely small number because for any value of x you choose, you can always use x+1 to generate an even smaller fraction.

=== **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.** === A limit affects the conditions for continuity because with a limit, it is plain and simple. With a limit, the graph is no longer continuous. It is a discontinued function.

Continuous Graphs: Discontinuous Graphs:

In these graphs the lines are continuous; there are no holes in In this graph above, the function is not continuous. To add to that, the graph and the line just keeps on going with no interruption there are also removable discontinuity. In addition, there are whatsoever. multiple limits in the graph. This means that there is no ONE SiNGLE limit. Thus, making it a discontinuous graph.