limits and graphs suppose that a function f has the graph shown on the right our goal is to describe the behavior of f in the vicinity of x equal to 1 in a concise manner let's first notice that the value F of one is equal to 1 yet if x is close but not equal to one then f ofx is close to two in in fact the closer X is to one the closer f ofx is to two so the number two is crucial in describing the behavior of f near one the way that we describe this behavior is to say that 2 is the limit of f ofx as X approaches one this is written compactly in the manner shown to be a little more precise the reason two is the limit as X approaches one is that for any interval centered at two on the Y AIS no matter how small the number F ofx will be in that interval for all X other than one in some sufficiently small interval centered at one on the x axis also we point out that the limit as X approaches one has nothing to do with the value of F at one we could change F of one to any number we like or even leave it undefined and the limit Remains Two note that if the limit as X approaches one is different from F of one there's a hole in the graph at the point 1 comma 2 if F of one were equal to the Limit the hole would be filled in fact at any point where the graph of f is continuous the y-coordinate that is the value of f will equal the limit of f as X approaches the x coordinate of that point so value and limit coincide wherever the graph of f is continuous this idea is the basis of the mathematical definition of continuity that you will see later let's look at another example again suppose that f is the function whose graph is shown on the right here the interesting behavior of the function is in the vicinity of x equals 0 Let's first notice that the value F of 0 is equal to 2 if x is close to and less than zero then f ofx is close to two in fact the closer X is to zero while X is less than zero the closer f ofx is to two but if x is close to and greater than zero then f ofx is close to one in fact the closer X is to zero while being greater than zero the closer f ofx is to one therefore there is no number that can serve as the limit of f ofx as X approaches zero that is the limit does not exist however we can describe the behavior of f near x equals 0 in terms of one-sided limits here 2 is the limit of f ofx as X approach approaches zero from the left or from below this means that for any interval centered at two on the Y AIS F ofx will be in that interval whenever X is in a sufficiently small interval whose right end point is zero one is the limit of f ofx as X approaches zero from the right or from above this means that for any interval centered at one on the Y AIS F ofx will be in that interval whenever X is in a sufficiently small open interval whose left end point is zero this example illustrates a very important fact about limits the limit of f ofx as X approaches some number a exists if and only if both of the one-sided limits as X approaches a exist and coincide that is if and only if F ofx approaches the same number as X approaches a from both the left and the right when this happens the limit equals the common value of the one-sided limits another example again suppose that f is the function whose graph is shown on the right here the interesting behavior of f is in the vicinity of xal 2 notice that F of 2 is undefined and the line x equals 2 is a vertical ASM toote if x is close to two while less than two then f ofx is large and positive in fact the closer X is to two while less than two the larger F of X is if x is close to two while greater than two then f ofx is large and negative therefore there is no number that can serve as the limit of f ofx as X approaches two that is the limit does not exist in fact neither of the one-sided limits as X approaches two exists however we can describe the behavior of the function near the vertical asmp toote in terms of infinite one-sided limits here we say that the limit of f ofx as X approaches 2 from the left is plus infinity and we say that the limit as X approaches 2 from the right is negative Infinity the limit from the left is positive Infinity because given any positive number no matter how large F ofx will be greater than that number for all X in a sufficiently small open interval whose right end point is two the limit from the right is negative Infinity because given any NE ative number no matter how large F ofx will be less than that number for all X in a sufficiently small open interval whose left end point is two we should point out that in this example since the one-sided limits do not agree the limit as X approaches two does not exist even in an infinite sense now let's look at a few example exercises this is the first one given the function graphed here we want to to determine a the value F of one B the limit of f ofx as X approaches one from the left C the limit of f ofx as X approaches one from the right and D the two-sided limit of f ofx as X approaches one because the point 1 one is on the graph we conclude that F of one is equal to 1 as X approaches two from the left the corresponding y-coordinates are function values are approaching two so the left-sided limit is two as X approaches one from the right corresponding function values are approaching zero so the right sided limit as X approaches 1 equals 0 now since the two one-sided limits do not agree the limit as X approaches one of f ofx does not exist this example is similar to the previous one given the function graph here we'd like to determine a the value of the function at two B the limit as X approaches two from the left C the limit as X approaches two from the right and D the limit as X approaches 2 F of 2 is equal to zero because the point 2 0 is on the graph as X approaches 2 from the left the corresponding y-coordinates or function values are approaching zero so the limit as X approaches 2 from the left equals 0 as X approaches two from the right function values are negative and getting larger and larger so the limit as X approaches two from the right is negative Infinity now because the two one-sided limits do not agree or simply because the right sided limit does not exist the limit is X approaches to of f ofx does not exist in this example exercise we want to sketch the graph of a function f defined for X greater than minus1 and less than three so that the list of conditions concerning limits Andor values at xal to minus1 0 1 2 and 3 are true let's begin at the left end point and work our way from left to right F of -1 is undefined and the right sided limit as X approaches -1 is 2 so near -1 the graph might look something like this since F of 0 is 1 we'll plot the point 01 and since the limit as X approaches 0 is 0 the graph might look something like this between -1 and 0 next we plot a point at 1 Z and we note that the infinite left-sided limit at 1 means that the line xal 1 is a vertical ASM toote since F ofx approaches positive Infinity as X approaches one from the left and again using the fact that the limit at zero is zero we conclude that the graph might look something like this between 0 and one next let's use the fact that the limit as X approaches one from the right is one to continue the graph past one the value at two and the left sided limit as X approaches two are both two and so we plot the point 22 and then sketch the graph for X between 1 and two something like this now since the right sided limit as X approaches 2 is zero we continue the graph past two like this and finally we finish the graph making use of the fact that the limit as X approaches three from the left is equal to 1 there are many other ways that this graph might have been drawn other possibilities need only show the correct values and limiting Behavior at x = -1 0 1 2 and 3 in particular this is another possibility where the graph Wiggles around a bit more between those values of x