he Definition of the Limit

he Definition of the Limit - he Definition of the Limit In...

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he Definition of the Limit In this section we’re going to be taking a look at the precise, mathematical definition of the three kinds of limits we looked at in this chapter. We’ll be looking at the precise definition of limits at finite points that have finite values, limits that are infinity and limits at infinity. We’ll also give the precise, mathematical definition of continuity. Let’s start this section out with the definition of a limit at a finite point that has a finite value. Definition 1 Let f(x) be a function defined on an interval that contains , except possibly at . Then we say that, if for every number there is some number such that Wow. That’s a mouth full. Now that it’s written down, just what does this mean? Let’s take a look at the following graph and let’s also assume that the limit does exist. What the definition is telling us is that for any number that we pick we can go to our graph and sketch two horizontal lines at and as shown on the graph above. Then somewhere out there in the world is another number , which we will need to determine, that will allow us to add in two vertical lines to our graph at and .
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Now, if we take any x in the pink region, i.e. between and , then this x will be closer to a than either of and . Or, If we now identify the point on the graph that our choice of x gives then this point on the graph will lie in the intersection of the pink and yellow region. This means that this function value f(x) will be closer to L than either of and . Or, So, if we take any value of x in the pink region then the graph for those values of x will lie in the yellow region. Notice that there are actually an infinite number of possible δ ’s that we can choose. In fact, if we go back and look at the graph above it looks like we could have taken a slightly larger δ and still gotten the graph from that pink region to be completely contained in the yellow region. Also, notice that as the definition points out we only need to make sure that the function is defined in some interval around but we don’t really care if it is defined at . Remember that limits do not care what is happening at the point, they only care what is happening around the point in question. Okay, now that we’ve gotten the definition out of the way and made an attempt to understand it let’s see how it’s actually used in practice. These are a little tricky sometimes and it can take a lot of practice to get good at these so don’t feel too bad if you don’t pick up on this stuff right away. We’re going to be looking a couple of examples that work out fairly easily. Example 1 Use the definition of the limit to prove the following limit. Solution
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he Definition of the Limit - he Definition of the Limit In...

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