 Hello and welcome to the session. In this session, we will discuss a question which says that determine whether the given functions are continuous at the given x value. First is f of x is equal to x square plus x minus 1. We will call x plus 3 given where x is equal to minus 3. And second is f of x is equal to x square plus 3x minus 1 given that x is equal to 2. Now before starting the solution of this question, we should know a result. And that is continuity at a point. Now a function is continuous at point x is equal to c. If it satisfies the three conditions, first is the function is defined at x is equal to c that is f of c exists. That is the function should not be undefined at x is equal to c. Second is the function approaches same y values on the left and y sides of x is equal to c. And third is the y value that a function approaches from each side is f of c. Now this result will work out as a key idea for solving the given question. Now let us start with the solution of the given question. Now in the first part we are given the function f of x is equal to x square plus x minus 6 over x plus 3. And we want to discuss its continuity as x is equal to minus 3. So using the key idea, we will check its continuity. Let us start with the first condition. And the first condition is the function should be defined at x is equal to c. So we are going to check whether this function is defined at x is equal to minus 3 or not. So the substitute x is equal to minus 3 in f of x. And we get f of x is equal to minus 3 whole square plus or minus 3 minus 6 whole upon minus and this implies is equal to minus whole upon minus 3 plus 3 which implies f of x is equal to upon 0 at 0 1 0 1 which is not defined. Continuity is not satisfied to minus 3. There is a whole or small gap in the graph. So this point is the point of discontinuity. But we are given the function f of x is equal to x square plus 3 x minus 1. And we have to discuss its continuity at x is equal to 2 now again. So we will check all these three conditions given in this continuity test. Whether the function is defined at x is equal to 2 or not. And we get is equal to 2 square plus 3 into 2 minus 1 which implies f of 2 is equal to plus 6 minus 1 and this gives f of 2 is equal to 10 minus 1 that is 9. But the given function is defined or in particular we have f of 2 is equal to 9. So the first condition of continuity is satisfied. Now we check the second condition of continuity. And the second condition is the function approaches y values on the left and right sides of x is equal to 2. We name table when x is less than 2 and x approaches to 2 table when x approaches to 2. Now we have made these two tables. In the first table we will take the values of x less than 2 and x approaches equal to 1.99, 1.99, 1.999. So we will put these values 1 by 1 and we get the corresponding values of y. Now using calculator go to 1.9 we get y is equal to 8.31. When x is equal to 1.99 we get y is equal to 8.9301 and for x is equal to 1.999 we get y is equal to 8.99301. So we will take the values of x greater than 2 is equal to 2.1, x is equal to 2.01 and x is equal to 2.001. And using calculator we get the corresponding values of y. Now the first table shows that when x is less than 2 and x approaches to 2, then y values approach 9 because all obtained values of f of x are very close to 9. Second table shows that when x is greater than 2 is 2 then y values approach 9 because here also all the values of f of x are very close to 9. Thus the functional approaches saying y value x is equal to 2 continuity is also satisfied. Now we will check the third condition of continuity. Now the third condition is the y value that the functional approaches from each side is f of 2. Now from these two tables we concluded that the y values approach 9 first condition 2 is equal to 9. Thus the y value that f of x approaches from each side is f of 2 that is 9. The given function is continuity x is equal to 2. Now this can also be confirmed by examining the graph of this function. Now see the graph of this function that there is no break in the graph. This is a continuous screen of the given question. That's all for this session. Hope you all have enjoyed this session.