LimitsIn order to introduce the notion of the limit, we will consider the following situation. Supposeyou are given a function, defined on some interval, except at a single point in the interval.What, if anything, can you say about the value of the function at that point?Let us suppose we are given the function f (x) = x2on [-1,0) (0,1] ie, the function is definedon [-1,1] except for at 0. What can we say about f(0)? Since we know the function is givenby x2at all points except for 0, it seems that the function should be defined in the sameway at 0. Thus, it seems like f(0) should be 0. However, a general function can be definedto have an arbitrary value at any point. Thus, both f (x) = x2on [-1,0) (0,1] with f (0) = 0and f(x) = x2on [-1,0) (0,1] with f (0) = 1 are both equally valid functions (as well asany other value for f (0)). Thus, despite whatever intuition we have about what value f(0)should have, there is nothing about our function that demands it have that value, so in factwe cannot conclude anything about the value of f(0).Despite the fact we can conclude nothing about the actual value of f(0), we want to buildon the notion that a function should have a certain function value at a c ertain point, whichis determined irrespective of the actual value at that point (ie, what f(x) actually is doesn’tdetermine what f(x) should be, in this notion of what should be). In this way, what afunction value should be at a given point is determined only by the values of the function atthe points around that point, but not the point itself.Just as we could say that a function should take on a specific value at a specific point basedon an algebraic definition of the function, we can also explore the notion of what should befrom a graphical standpoint. If we graph the function f(x) = x2on [-1,0) (0,1] it looks like acontinuous function with a hole in the middle of it. The value that we previously determinedshould be f(0) is exactly the one which fills the hole. Thus, in some sense the function valuethat should be defined for f (0) is the one that fits in with the graph. Given that we candefine a function in any way we like, it is not difficult to c onstruct a function for which thereis no value that ‘fills in the hole.’ It is worth noting, however, that if there is a value whichfills the hole, it will be unique. When we build the limit concept from this, it will have bothof these important properties - it will not necessarily exist, but if it does, it will be unique.The limit is in some sense a refined version of the above concept. Intuitively, the notion ofa limit is the value a function approaches as its inputs approach a given point. Thus, whenwe writelimx→x0f(x) = Lwe mean that as the values x in the domain of the function approach the point x0, thefunction values f(x) correspondingly approach the value L. As previously stated, the actualvalue of a function at a point does not determine in any way what limit (if any) the functionapproaches (the function need not even be defined at the point in question). This is consistentwith our previous notion of the function value that seemed like it should be the value of thefunction at the given point. Thus, for our previous example we would find thatlimx→0f(x) = 0(the limit as x approaches 0 for our previous function is 0). It is important to emphasizethat the value of a limit is not just a value a function approaches at the given point, but avalue it approaches in a very special way (surely this must be so, as an increasing functionapproaches all larger values, but not all larger values seem like they should be the right valuefor the function at a given point). The key to describing this special way that the functionvalues approach the limit value lies in approximation. Consider the following definition: wesay thatlimx→x0f(x) = Lif for any error tolerance around L, we can restrict the values of f (x) to be within that errortolerance of L by restricting the values of x to be sufficiently close to x0; ie. for all values ofx within some interval around x0(excluding x0) the distance between f (x) and L is withinthe error tolerance. Note that this requires we be able to find such an interval no matterhow small the error tolerance is (but an error tolerance must be greater than 0, because weare still considering an approximation, just an approximation we can make as accurate aswe like); in general, a tighter error tolerance will necessitate a smaller interval around thepoint x0we are interested in. Such an interval cannot contain even a single point for whichthe function value lies out of the error tolerance around L. While we must be able to findan interval for any error tolerance to show a function approaches a limit at some point, weneed only to find a single error tolerance for which there is no such interval in order to showa function does not approach a specific limit L at some point (and we need to show this forall values L in order to show there is no limit). Finally, it follows from this definition thata limit is unique; that is, a function can only approach one value at a given point in thisspecial way, if it approaches any value at all.This refined definition is very important, because most cases will not be as simple as thefunction we previously considered. We will revisit this notion later with the formal definitionof a limit, but before doing so we will attempt to gain further insight into limits using theless daunting, intuitive notion. Our approach will be to begin with very simple functions,and use our understanding of them to calculate limits of more complicated functions.Before doing so, it should be emphasized that a function need not have a limit at a givenpoint. Consider the function f (x) given byf(x) =|x|xwhich is defined for all x, with x 6= 0. This function does not have a limit as x approaches 0.Inuititively, the graph is broken with a jump in it, so there is no way for the function valuesto be approaching a single value at x = 0. The way to see this based on our definition is asfollows: for any interval around 0 (an interval must be more than a single point), we can findan x1with x1< 0 and an x2with x2> 0. Thus, f (x1) = −1 and f (x2) = 1. For any errortolerance less than 1, it is impossible to find a value L so that both of these function values(-1 and 1) are within the error tolerance around it, yet any interval around 0 must containsuch values x1and x2. Thus, this function has no limit at x = 0. There