 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 raise to power x plus 2 raise to power x plus 3 raise to power x plus 4 raise to power x upon 4 whole raise to power 3 by x. Now in Hopeter's rule states that 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 dash of x. With this key idea let us proceed with the solution. Now we have to find the value of the expression limit x tends to 0, 1 raise to power x plus 2 raise to power x plus 3 raise to power x plus 4 raise to power x whole raise to power 3 by x is equal to 1 raise to power x plus 2 raise to power x plus 3 raise to power x plus 4 raise to power x whole upon 4 raise to power 3 by x taking log on both the sides log of y is equal to 3 by x into log of 1 raise to power x plus 2 raise to power x plus 3 raise to power x plus 4 raise to power x whole upon 4 which implies that log of y is equal to 3 by x log of 1 raise to power x plus 2 raise to power x plus 3 raise to power x plus 4 raise to power x minus of log of 4 since log of a upon b is equal to log of a minus log of b and this can be written as log of y is equal to 3 into log of 1 raise to power x plus 2 raise to power x plus 3 raise to power x plus 4 raise to power x minus log of 4 whole upon x. Now taking the limit on both the sides we have limit x tends to 0 log of y is equal to 3 into limit x tends to 0 log of 1 raise to power x plus 2 raise to power x plus 3 raise to power x plus 4 raise to power x minus of log of 4 whole upon x. Now if we put the value of x as 0 in this expression we get so we have log of 1 raise to power 0 plus 2 raise to power 0 plus 3 raise to power 0 plus 4 raise to power 0 minus log of 4 whole upon 0 which is equal to log of 1 raise to power 0 is 1 plus 2 raise to power 0 is 1 plus 3 raise to power 0 is 1 plus 4 raise to power 0 is 1 minus log of 4 upon 0 which is equal to log of 4 minus log of 4 whole upon 0 which is of 0 by 0 form. So, expression on the right hand side is of 0 by 0 form according to L Hoppeter's rule 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 upon g of x is equal to limit x tends to a f dash of x upon g dash of x therefore applying L Hoppeter's rule on the right hand side of the equation we get 3 into limit x tends to 0 differential of the numerator with respect to x that is differential of log of 1 raise to power x plus 2 raise to power x plus 3 raise to power x plus 4 raise to power x with respect to x we have 1 upon 1 raise to power x plus 2 raise to power x plus 3 raise to power x plus 4 raise to power x into differential of 1 raise to power x plus 2 raise to power x plus 3 raise to power x plus 4 raise to power x with respect to x i.e. 1 raised to power x into log of 1 plus 2 raised to power x into log of 2 plus 3 raised to power x into log of 3 plus 4 raised to power x into log of 4 minus differential of log of 4 with respect to x i.e. 0 upon differential of x with respect to x i.e. 1. Now putting the value of x i 0 we get 3 into 1 raised to power 0 log of 1 plus 2 raised to power 0 log of 2 plus 3 raised to power 0 log of 3 plus 4 raised to power 0 log of 4 upon 1 raised to power 0 plus 2 raised to power 0 plus 3 raised to power 0 plus 4 raised to power 0 which is equal to 3 into log of 1 plus log of 2 plus log of 3 plus log of 4 upon 1 plus 1 plus 1 that is 3 upon 4 into log of 1 plus log of 2 plus log of 3 plus log of 4 which can be written as 3 by 4 into log of 1 into 2 into 3 into 4 since log of a plus log of b plus log of b can also be written as log of a b c b which is equal to 3 by 4 into log of 24. So we have limit x tends to 0 log of y is equal to 3 by 4 into log of 24 which can also be written as log of limit x tends to 0 y is equal to log of 24 raised to power 3 by 4 which implies that limit x tends to 0 y is equal to 24 raised to power 3 by 4 and y is given by 1 raised to power x plus 2 raised to power x plus 3 raised to power x plus 4 raised to power x whole upon 4 whole raised to power 3 by x therefore we have limit x tends to 0 1 raised to power x plus 2 raised to power x plus 3 raised to power x plus 4 raised to power x upon 4 whole raised to power 3 by x is equal to 24 raised to power 3 by 4 which is the required answer. This completes our session hope you enjoyed this session.