consider the pwi function f ofx and let's say that it's equal to 5x + 3 when X is less than 2 and it's equal to 2x^2 + 5 when X is between 2 and 4 and then it = X Cub - 5x + 3 when X is equal to or greater than 4 so what is the limit as X approaches 2 from the left side of f ofx so which portion of the pie wise function should we use now as we approach two from the left side should we choose a value like 1.99 or 2.01 on a number line here's two here's 1.99 and here's 2.01 so as weach approach two from the left side we have to pick a value that's less than two ideally 1.99 so therefore we need to use 5x + 3 because that's when X is less than two that's the left side of x = 2 so it's going to be 5 * 2 2 + 3 which is 10 + 3 so that's equal to 13 now what is the limit as X approaches 2 from the right of f ofx on the right side that is greater than 2 we need to use this portion of the peie wise function 2x^2 + 5 so it's 2 * 2^ 2 + 5 2^ 2 is 4 2 * 4 is 8 8 + 5 is 13 now let's find the limit as X approaches two from either side so because these two are the same you know that it's going to be 13 now here's the next question what is the value of f of two which function or which portion of the piecewise function should we use to find the value of f of two so X doesn't equal two for this particular portion of the function but it does equal two in this function so we're going to have to use 2x^2 + 5 which we already know what the answer is this is equal to 13 so F of 2 is defined now what is the limit as X approaches four from the left side so from the left side of four that is when X is less than 4 we need to use 2x^2 + 5 so it's going to be 2 * 4^ 2 + 5 4^ 2 is 16 2 * 16 is 32 32 + 5 well that's equal to 37 now what is the value of the limit as X approaches 4 from the right side this time x has to be greater than four so we need to use this portion of the peie wise function so let's plug in four 4 the 3r 4 * 4 is 16 * 4 that's 64 5 * 4 is 20 and 64 - 20 is 44 and 44 + 3 that's 47 so even though the one-sided limits exists because they don't match the limit itself does not exist so the limit as X approaches four from either side does not exist in this particular uh problem now there's one more question that we need to answer what is the value of f of four so when X is four what is the value of the function X is four using this portion of the P wise function and so as we mentioned before this is going to be 47 so let's work on another example so here's another peie wise function when X is less than 1 1 we have the portion 7X - 5 and then the function is going to be equal to 3x^2 - x when X is greater than 1 but less than or equal to three now the function will have a value of five when X is equal to 1 and it's going to be X Cub + 4 when X is greater than 3 so using this information what is the limit as X approaches pos1 from the left side so one from the left side that's less than one so therefore we have to use 7X - 5 so that's going to be 7 * 1 - 5 7 - 5 is 2 so that's the value of the left-sided limit now what about the right side limit as X approaches one from the right side what is the value of f ofx so this time we need to use 3x^ 2 - x 1^ 2 is 1 * 3 that's 3 - one that's going to be equal to two so because these two the left side the limit and the right side limit because they're the same the limit as X approaches one from either side will also be equal to two now what is the value of f of one so when X is exactly one what is the value of the function so when X is one notice that the value of the function is five so F of 1 is equal to 5 now what is the limit at as X approaches three from the left side so that's when X is less than three so we got to use 3x^ 2 - x so let's plug in three it's going to be 3 * 3^ 2 - 3 3^ 2 is 9 * 3 that's 27 27 - 3 is 24 now what about the right sided limit what is the limit as X approaches 3 from the right side so when X is greater than 3 we need to use this portion of the equation so X Cub + 4 so it's going to be 3 3r + 4 that's uh 27 + 4 which is 31 so the left side limit and the right side limit there's a mismatch they don't equal the same thing they're therefore the limit as X approaches 3 from either side we could say does not exist now what is the value of f of3 to find the value of f of three we need to use this portion because X is less than and equal to 3 when we plug in 3 into 3x^2 - x we got 24 so F of 3 is 24 consider the function f ofx which is equal to CX + 3 when X is less than 2 and that's equal to 3x + C when X is equal to or greater than 2 What is the value of the constant C that will make the function continuous at xal 2 so what do we need to do in order to find the value of that constant C well if it's it's going to be continuous at xal 2 these two functions have to have the same y value which means that they must be equal to each other so the first thing is to set them equal to each other and they have to be equal to each other at an x value of two so the second step is to replace x with two and then find the value of C so we're going to have 2 C + 3 is equal to 6 + C so let's subtract both sides by C and let's subtract both sides by three 2 c - c is C 6 - 3 is 3 so C is equal to 3