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Not all functions have limits at all points, and we discuss what this means and how we can tell if a function does or does not have a limit at a particular value. In this chapter we will discuss just what a limit tells us about a function as well as how they can be used to get the rate of change of a function as well as the slope of the line tangent to the graph of a function (although we'll be seeing other, easier, ways of doing these later). For a continuous function, evaluating a limit requires just evaluating the function (once you know it really is continuous)
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It just happens, as we’ll see, that the primary purpose of limits in calculus, the derivative, is a case where you can’t (just) plug in a number In fact, early mathematicians used a limiting process to obtain better and better approximations of areas of circles. Actually, limits are the basis for calculus
They are not so used directly in practice (by practice i mean other subjects, such as physics), but the concepts that are defined using them (pretty much entire calculus) are widely used
This caters for the limit to zero What about the limit to infinity Our prediction, the limit, isn’t required to match reality But for most natural phenomena, it sure seems to
Limits let us ask “what if?” If we can directly observe a function at a value (like x=0, or x growing infinitely), we don’t need a prediction. Evaluating the limit of a function at a point or evaluating the limit of a function from the right and left at a point helps us to characterize the behavior of a function around a given value. Are limits only for difficult functions
Limits can be used even when we know the value when we get there
Nobody said they are only for difficult functions We know perfectly well that 10/2 = 5, but limits can still be used (if we want!) infinity is a very special idea. For many applications, it is easier to use the definition to prove some basic properties of limits and to use those properties to answer straightforward questions involving limits The most important properties of limits are the algebraic properties, which say essentially that limits respect algebraic operations:
Limit (mathematics) in mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value [1] limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit or limiting process, essential to the understanding of calculus, has been around for thousands of years
