 Hello and welcome to the session. In this session, we will discuss the continuity and end behavior of a function by seeing their graphs. Now we have already discussed graphs of linear functions and quadratic functions. We see that the graphs of these functions are continuous. It means there is no breakage or gap in the curves of these functions. But sometimes we have graphs of functions which are discontinuous. That is, there exists a gap or breakage in the curve of the function. Now seeing this graph, there are gaps or breaks in the curve. Here we cannot draw it without lifting the pencil and this is the graph of discontinuous function. That is, we cannot draw the graph of the function without lifting the pencil. Now there are many types of discontinuities. First is, if I write discontinuity in this, the unit value of f of x becomes greater and greater as graph approaches given value of x. Now here, the function f of x is equal to 1 upon x minus 1. Here we can see a small change in x near x is equal to 1 on both sides. That is, left and right of x is equal to 1 gives a very large change in the value of the function continuity as x is equal to 1. Next is, jump discontinuity in this. The graph stops at a given value of the domain and then begins again at different range value for the same value of the domain. Now suppose given the function f of x is equal to 1, if x is less than 0, x is greater than equal to 0. Now let us see graph of this function. Now see the graph of this function. Here you can see the graph stops at x is equal to 0 and then stops again from x is equal to 0. Watch at different value of y. Here you can see y is equal to 3, y is equal to 1. So this is called jump discontinuity. Next is, remove away discontinuity. Now, remove away discontinuity. If the function is continuous at 1 to c, the value of the function becomes and whole appears at the point of discontinuity. Now see the graph of this function. Where the function is continuous at all the domains except at x is equal to 2, this whole appears at the point of discontinuity that is at x is equal to 2. Now, function is not discontinuous. Then it is said to be continuous that is said to be continuous at x is equal to c. If graph of function passes through that point without a break and so you must note that linear, quadratic, exponential, logarithmic and constant functions are continuous at all points. Now, let us discuss continuity at a point. Now a function is continuous at point x is equal to c. If x satisfies the three conditions, the function is defined at x is equal to c that is f of c exists which means the function should not be undefined at the point x is equal to c. Second is the function approaches same y values on the left and right sides of x is equal to c, limit of the function f of x where x tends to c exists and third is the y value that a function approaches from each side is that is limit of function f of x when x tends to c is equal to f of c, function f of x is continuous on an interval. If only if it is continuous, now we will discuss ant behavior of polynomial. Now the ant behavior of a function describes what the y values do when x becomes greater and greater. When x becomes greater and greater we see that x approaches to infinity by more negative than with y and with real numbers instead of infinity. Now in polynomials the ant behavior is found by searching value of f of x when absolute value of x becomes greater and greater. Now let us consider the following example. Here we have the function f of x is equal to x cube minus x square minus x plus 1. Now we will analyze its ant behavior both graphically and numerically. Now using graphing calculator we get the following graph of the function. To observe on both ends of the curve the value of the function becomes greater and greater with increase in value of x. Now on this side you can see this curve approaches to plus infinity and the curve approaches to negative infinity when x approaches to negative infinity that limit of the function f of x when x tends to infinity is equal to limit of the function f of x when x tends to minus infinity is equal to minus infinity. Now let us analyze it numerically. Now using calculator we construct the table of values. Option values x increases. Now when we examine ant behavior we usually take from greater the value of f of x also becomes greater and greater is to plus and minus infinity in general. Here at the polynomial function can be modeled by a function comprising solely of the term comprising highest power of x equal to 0 is equal to minus 1 into x raise to power n minus 1 plus so on plus a done ant behavior. Option p of x in which we have discussed earlier f of x is equal to x cube minus x square minus x plus 1 will have same ant behavior x cube. So this function we have same ant behavior as the function containing the single term ant behavior of polynomials can be summarized in following manner. The function p of x is equal to 2x square where is positive infinity infinity. Now if a n is negative is even for example see the graph of the function p of x is equal to minus where a n is negative and n is e is to minus infinity infinity is to minus infinity graph of the function p of x is equal to minus 2 is negative and n is when in this case minus infinity we have discussed ant behavior of a function and this completes the session hope you all have enjoyed the session.