 Hello and welcome to the session. In this session we discuss the following question that says Evaluate the limit x times 2a sin x minus sin a is drawn upon root x minus root a. Before we move on to the solution, let's recall some formula. We have sin c minus sin d is equal to 2 cos c plus d by 2 into sin c minus d by 2. The other formula is cos of a plus b and this is equal to cos a into cos b minus into sin b. We have a standard result of trigonometric limits which is limit x times 2 0 sin x upon x equal to 1. This is the key idea that we use for discussion. Let's proceed with the solution now. We are supposed to find the limit. The limit x times 2 and this will upon root x we will put x equal to supporting x equal to a plus s. We get that. Therefore this limit would be equal to limit in the numerator and the denominator we will put a plus h. So this would be minus root a. Now for the numerator of this function we can use the formula sin c minus sin d plus d by 2 into sin c minus d by 2. So using this formula for the numerator of this function we get this is equal to limit h times 2 0 minus a this will upon square root of a plus h minus square root of. Now to rationalize this denominator limit h times 2 0 a plus h plus square root of a and this will upon this is equal to limit h times 2 0. Now we can h by 2 into sin h by 2 into square root of a plus h plus this is equal to limit h times 2 0 cos of a plus h by 2 into limit h times 2 0 by 2. This will upon h into limit h times 2 0 square root of a plus h plus square root of this function that is cos of a plus. We can use the formula cos of a plus b is equal to cos a cos p minus sin a sin v times 2 0 into cos h by 2 minus here we can have limit h upon 2 times 2 0 upon 2. This will upon h upon 2 into limit h times 2 0 square root of a plus h plus square root of this limit we can put h equal to 0 and we get this is equal to into cos 0 which is 1 minus into sin 0 which is 0. This will into times 2 0 sin h upon 2 this will upon h upon 2 standard result of trigonometric limit which is limit x times 2 0 sin x upon x is 1. So this into 1 h equal to 0 here we get root a plus root a which is 2 root so this is equal to if we understood the solution of this question.