 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says proof that the function fx equal to 5x minus 3 is continuous at x equal to 0, at x equal to minus 3 and at x equal to 5. First of all let us understand what is a continuous function. If we can draw the graph of the function without lifting the pen from the plane of the paper, then the function is said to be continuous or we can say function f is continuous at x equal to a. If function f is defined at x equal to a or we can say f a exists and limit of the function is equal to value of the function at x equal to a. Now let us start the solution. Function f is given by fx equal to 5x minus 3. Now let us take the continuity of the function at x equal to 0. We know this is a polynomial function and polynomial function is defined at every real number. So given function is defined at x equal to 0. So this satisfies the first addition of the key earlier. Now let us check if the limit of the function is equal to value of the function at x equal to 0. Now we will find limit of x tending to 0 fx is equal to limit of x tending to 0 5x minus 3. Now this is equal to 5 multiplied by 0 minus 3 or we can simply write it as minus 3. We know 5 multiplied by 0 is 0. So 0 minus 3 is equal to minus 3. Now limit of the function at x equal to 0 is minus 3. Now let us find out value of the function at x equal to 0. Now f 0 is equal to 5 multiplied by 0 minus 3 which is equal to minus 3. Now clearly we can see limit of the function is equal to value of the function at x equal to 0. So given function f is continuous at x equal to 0. Now this expression satisfies the second condition of the key earlier. So given function f is continuous at x equal to 0. Now let us check if the given function f is continuous at x equal to minus 3. We know at x equal to minus 3 function f is defined. Now let us find out limit of the function at x equal to minus 3 limit of x tending to minus 3 fx is equal to limit of x tending to minus 3 by x minus 3. Now this is equal to 5 multiplied by minus 3 minus 3. Simplifying we get minus 15 minus 3. Now this is equal to minus 18. So we get limit of the function at x equal to minus 3 as minus 18. Now let us find out value of the function at x equal to minus 3. f minus 3 is equal to 5 multiplied by minus 3 minus 3 which is equal to minus 15 minus 3. Simplifying we get minus 18. Now value of the function at x equal to minus 3 is minus 18 and limit of the function at x equal to minus 3 is also minus 18. So given function is continuous at x equal to minus 3. Now let us take the continuity of the function at x equal to 5 at x equal to 5. Function f is defined. Now let us find out limit of the function at x equal to 5. So we can write limit of x tending to 5 fx is equal to limit of x tending to 5 5x minus 3. Now this is equal to 5 multiplied by 5 minus 3. Simplifying we get 22. So limit of the function at x equal to 5 is 22. Now let us find out value of the function at x equal to 5. This is equal to 5 multiplied by 5 minus 3 which is further equal to 22. Now clearly we can see limit of the function is equal to value of the function at x equal to 5. So this implies given function f is continuous at x equal to 5. So given function f is continuous at x equal to 0, x equal to minus 3 and x equal to 5. Hence proved this complete session.