![]() ![]() But now we'd like to use the formal definition of the limit to better understand why the limit exists. We have already calculated this limit both graphically and algebraically and determined that it is 2. ![]() A simple example, where lim x → c f(x) = f(c):įor this function, we are interested in the limit as x approaches 1: We do this both by using graphs to see if we can approximate the appropriate values for δ or M, and by seeing if we can calculate these values exactly by approaching the equation algebraically. Limits of Specific Functionsįor each of the following examples, we look at how the formal definition of the limit allows us to prove that the limit exists or that it does not exist. In cases where the limit does not exist, we should be able to see why a δ won't exist for every possible ε: in other words, we should be able to find an ε in these cases for which no possible δ can be found which will force f(x) to remain within a distance ε of L. Let's see how this definition can be applied to the example limit calculations that we've done in previous lectures. So, if a limit exists, it should be possible to bound an area around c that will force f(x) to stay within any chosen specific distance of L. This is also probably best understood by looking at a graph: ![]() In other words, if we pick an interval on the y-axis around L, we can always find a cutoff value on the x-axis that will force f(x) to stay with the chosen range of y-values once it is past this cutoff location. If f(x) is a function and L is a real number, thenįor any number ε>0 that we choose, it is possible to find another number M>0 so that:įor all x's greater than M, f(x) will fall between L-ε and L+ε. We can also use this same idea to create a definition for limits at infinity: This is probably best understood by looking at a graph: In other words, if we pick a value on the y-axis around, we can always find an interval on the x-axis around c that will force f(x) to stay above this value (except for perhaps at f(c)). If f(x) is a function that is defined on an open interval around x=c, thenįor any number M>0 that we choose, it is possible to find another number δ>0 so that:įor all x's between c-δ and c+δ (except possibly at c exactly), f(x) will be greater than M. If we want a formal defintion of what it means for a limit to increase or decrease without bound, we can also adapt this approach to this case: In other words, if we pick an interval on the y-axis around L, we can always find an interval on the x-axis around c that will force f(x) to stay with the chosen range of y-values (except for perhaps at f(c)). If f(x) is a function that is defined on an open interval around x=c, and L is a real number, thenįor any number ε>0 that we choose, it is possible to find another number δ>0 so that:įor all x's between c-δ and c+δ (except possibly at c exactly), f(x) will fall between L-ε and L+ε. We do this now by providing a formal mathematical definition: In order to come up with a formal definition, we will need to clarify exactly when we can say that x or f(x) approach a specific value. However, this definition is informal because we haven't formally defined what we mean by "approaches" or "eventually gets closer and closer to". Lim x → c f(x) = L to denote "the limit of f(x) as x approaches c is L" Lim x → c+ f(x) = L to denote "the limit of f(x) as x approaches c from the left is L" Lim x → c- f(x) = L to denote "the limit of f(x) as x approaches c from the left is L" ![]() If f(x) has different right and left limits, then the two-sided limit ( lim x → c f(x)) does not exist. If f(x) never approaches a specific finite value as x approaches c, then we say that the limit does not exist. If the limit of f(x) as x approaches c is the same from both the right and the left, then we say that the limit of f(x) as x approaches c is L. If f(x) eventually gets closer and closer to a specific value L as x approaches a chosen value c from the left, then we say that the limit of f(x) as x approaches c from the left is L. If f(x) eventually gets closer and closer to a specific value L as x approaches a chosen value c from the right, then we say that the limit of f(x) as x approaches c from the right is L. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |