 Hello and welcome to the session. In this session, we are going to discuss the following question which says that Evaluate A limit 1 minus kushekov x into kushekov x as x tends to pi by 2 B limit 1 by 2 minus x into kushekov pi x as x tends to 1 by 2 In Hopitian's rule states that if f of x and g of x are functions such that f of a is equal to 0 and g of a is equal to 0 then limit f of x by g of x as x tends to a is equal to limit f of x by g of x as x tends to a With this key idea, we shall proceed with the solution. We are to find the value of the expression limit 1 minus kushekov x into kushekov x by 2 Now if we put the value of x as pi by 2 in this expression, we get 1 minus kushekov pi by 2 into kushekov pi by 2 kushekov pi by 2 is 1, therefore it becomes 1 minus 1 that is 0 into kushekov pi by 2 is undefined Therefore it takes 0 into infinity form and this can also be written as limit 1 minus kushekov x into kushekov x can be written as 1 upon kushekov x as x tends to pi by 2 Now again if we put the value of x as pi by 2, the expression becomes 1 minus kushekov pi by 2 upon kushekov pi by 2 Since kushekov pi by 2 is 1, therefore the numerator becomes 1 minus 1 that is 0 and in the denominator we have kushekov pi by 2 which is 0 So this is of 0 by 0 form and according to n-hospital's rule we have if f of x and g of x are functions such that f of a is equal to 0 and g of a is 0 then limit f of x upon g of x as x tends to a is equal to the limit f dash of x upon g dash of x as x tends to a Therefore applying the n-hospital's rule we have limit differential of the numerator with respect to x that is differential of 1 minus kushekov x with respect to x we have Differential of 1 with respect to x is 0 minus of differential of kushekov x with respect to x is minus of quarter of x into kushekov x upon differential of course of x with respect to x that is minus of sin of x as x tends to pi by 2 and this can be written as limit quarter of x into kushekov x by minus of sin of x as x tends to pi by 2 Now if we put the value of x as pi by 2 in this expression it becomes quarter of pi by 2 into kushekov pi by 2 whole upon minus of sin of pi by 2 which is equal to quarter of pi by 2 is 0 into kushekov pi by 2 is 1 upon minus of sin of pi by 2 which is 1 this is equal to 0 upon minus 1 that is 0 Therefore the value of the expression limit of 1 minus kushekov x into kushekov x as x tends to pi by 2 is equal to 0 Which is the required answer Now we shall find the value of the expression limit 1 by 2 minus x into kushekov pi x as x tends to 1 by 2 Now if we put the value of x as 1 by 2 in this expression it becomes 1 by 2 minus of 1 by 2 into kushekov pi by 2 So we have 1 by 2 minus 1 by 2 that is 0 into kushekov pi by 2 which is not defined So this is your 0 into infinity form and this can also be written as limit 1 by 2 minus of x kushekov pi x can be written as 1 upon kushekov pi x as x tends to 1 by 2 Now if we put the value of x as 1 by 2 in this expression it becomes 1 by 2 minus 1 by 2 that is 0 upon kushekov pi by 2 that is also 0 So this is your 0 by 0 form and according to L-Hopit's rule we have if f of x and g of x are the two functions such that f of a is 0 and g of a is 0 then limit f of x upon g of x as x tends to a is equal to limit f dash of x upon g dash of x as x tends to a Therefore applying L-Hopit's rule we have limit differential of the numerator that is 1 by 2 minus x with respect to x that is minus 1 upon differential of the denominator with respect to x that is differential of cos of pi x with respect to x which is minus of sin of pi x into differential of pi x with respect to x that is pi as x tends to 1 by 2 and this can be written as limit 1 upon pi into sin of pi x x tends to 1 by 2 Now putting the value of x as 1 by 2 in the expression we get 1 upon pi into sin of pi by 2 which is equal to 1 upon pi into sin of pi by 2 is 1 that is 1 upon pi Therefore the value of the expression limit 1 by 2 minus x into sin of pi x as x tends to 1 by 2 is equal to 1 upon pi which is the required answer this completes our session hope you enjoyed this session