 Hi, and welcome to our session. Let us discuss the following question. The question says, evaluate the following limits in exercises 1 to 22. Limit of cos 2x minus 1 by cos x minus 1 as x tends to 0. Let's now begin with the solution. In this question, we have to evaluate limit of cos 2x minus 1 by cos x minus 1 as x tends to 0. We know the limit of a technometric function is the value of that function at the point to which it is approaching. Now here, the point is 0. Now, in putting x as 0, in this, we get numerator as 0 and also denominator as 0. And we have learned that if limit takes the form 0 by 0, then we have to cancel the common factors from both numerator and denominator. So let's now first reduce this expression into lowest form. Now here, the limit takes the form 0 by 0. And we have learned that if the limit takes the form 0 by 0, then we cancel the common factors from both numerator and denominator. So let's now first reduce this into the lowest terms. So this is equal to limit x tends to 0. We know that 1 minus cos 2x is equal to 2 sine square x. This means cos 2x minus 1 is equal to minus 2 sine square x. But taking minus common from the denominator, we get minus into 1 minus cos x. Now cancel minus from both numerator and denominator. So we are left with limit x tends to 0, 2 sine square x by 1 minus cos x. Now sine square x is equal to 1 minus cos square x. So this is equal to limit x tends to 0, 2 into 1 minus cos square x by 1 minus cos x. This is equal to limit x tends to 0, 2 into 1 plus cos x. Into 1 minus cos x, we have used the identity of a square minus b square upon 1 minus cos x. Now cancel 1 minus cos x from both numerator and denominator. So we are left with limit x tends to 0, 2 into 1 plus cos x. And this is equal to 2 into 1 plus cos 0, cos 0 is equal to 1. So we have 2 into 2. And this is equal to 4. So our required answer is 4. So this completes the session. Bye and take care.