 Hello and welcome to the session. In this session we are going to discuss the following question which says that evaluate limit x tends to 0 1 minus x raise to power 1 by x n-hopitance rule states that if f of x and g of x are the two functions such that f of a is equal to 0 and g of a is equal to 0 then limit x tends to a f of x by g of x is equal to limit x tends to a f dash of x upon g with this key idea we shall proceed with the solution we are to find the value of the expression limit x tends to 0 1 minus x raise to power 1 by x if we put the value of x as 0 we get 1 minus 0 raise to power 1 by 0 that is of 1 raise to power infinity form now let y be equal to 1 minus x raise to power 1 by x taking log on both the sides we get log of y is equal to 1 by x into log of 1 minus x or we can write it as log of y is equal to log of 1 minus x upon x now taking limit on both the sides we have limit x tends to 0 log of y is equal to limit x tends to 0 log of 1 minus x by x now if we put the value of x as 0 in this expression we get log of 1 minus 0 upon 0 that is log of 1 upon 0 which is of 0 by 0 form since log of 1 is 0 now according to n-hopitans rule we have if f of x and g of x are the functions such that f of a is equal to 0 and g of a is equal to 0 then limit x tends to a f of x upon g of x is equal to limit x tends to a f dash of x upon g dash of x now applying n-hopitans rule on the right hand side of the equation we get limit x tends to 0 log of y is equal to limit x tends to 0 differential of the numerator with respect to x that is differential of log of 1 minus x with respect to x which is equal to 1 upon 1 minus x into differential of 1 minus x with respect to x that is minus 1 upon differential of x with respect to x that is 1 which is equal to limit x tends to 0 minus 1 upon 1 minus x now putting the value of x as 0 in this expression we get minus 1 upon 1 minus 0 that is equal to minus 1 so limit x to 0 log of y is equal to minus 1 or we can write it as log of limit x tends to 0 y is equal to minus 1 which implies that limit x tends to 0 y is equal to e raise to power minus 1 and we have assumed the value of y as 1 minus x raise to power 1 by x so we write limit x tends to 0 1 minus x raise to power 1 by x is equal to 1 by e and therefore the value of the expression limit x tends to 0 therefore 1 minus x raise to power 1 by x is equal to 1 upon e which is the required answer this completes our session hope you enjoyed this session