 Hi and welcome to our session. Let us discuss the following question. The question says evaluate the following limits and exercises 1 to 22. Limit z tends to 1 z to the power 1 by 3 minus 1 by z to the power 1 by 6 minus 1. Before solving this question, we should first we will verse with theorem 2 given in your NCEERT book. There are two states that for any positive integer n limit x tends to a x to the power n minus a to the power n upon x minus a is equal to n into a to the power n minus 1. This theorem holds n is a rational number. The knowledge of this theorem is the key idea in this question. Let us now begin with the solution. In this question, we have to evaluate limit z tends to 1 z to the power 1 by 3 minus 1 upon z to the power 1 by 6 minus 1. Now in order to make use of this theorem, we have to multiply and divide this expression by z minus 1. So on multiplying and dividing this expression by z minus 1, we get limit z tends to 1 z to the power 1 by 3 minus 1 upon z minus 1 into z minus 1 upon z to the power 1 by 6 minus 1. We can write this expression as limit z tends to 1 z to the power 1 by 3 minus 1 upon z minus 1 divided by limit z tends to 1 z to the power 1 by 6 minus 1 upon z minus 1. Now this can further be written as limit z tends to 1 z to the power 1 by 3 minus we can write 1 as 1 to the power 1 by 3 upon z minus 1 divided by limit z tends to 1 z to the power 1 by 6 minus we can write 1 as 1 to the power 1 by 6 upon z minus 1. We have learnt that limit x tends to a x to the power n minus a to the power n upon x minus a is equal to n into a to the power n minus 1. Now here in place of x we have z and in place of n we have 1 by 3 and in place of a we have 1. So limit of this expression is equal to n that is 1 by 3 into a that is 1 to the power n minus 1 that is 1 by 3 minus 1 divided by now here in place of n we have 1 by 6 and in place of a we have 1 and in place of x we have z. So limit of this expression is 1 by 6 into 1 to the power 1 by 6 minus 1. This is equal to 1 by 3 divided by 1 by 6 and this is equal to 1 by 3 into 6 and this is equal to 2. Hence our required limit is 2. So this completes the session by intake care.