 Hello and welcome to the session. In this session we discuss the following question which says evaluate the limit limit x tends to 0 cos ax minus cos dx this will be upon x square. Before we move on to the solution let's recall one result which says cos c minus cos d is equal to 2 sin plus d by 2 into we have a standard result of trigonometric limits which is limit x tends to 0 then x is equal to 1. This is the key idea that we use for this question. Let's proceed with the solution now. We are supposed to find the limit limit x tends to 0 cos ax minus cos dx and this will upon x square. Now for this numerator of the function that is cos ax minus cos dx we can easily apply this formula of cos c minus cos d. So using this formula we get limit x tends to 0 2 sin plus b by 2 into by 2 into x. Now taking this 2 outside this limit so this is equal to 2 into limit x tends to 0 a plus b by 2 x into 2 upon x square. Now we can write this x square becomes of a plus b by tan b minus a by 2 into x. Now we can write this this x square could be written as a plus b by 2 into 2 by a plus b into by 2 into x into 2 by b minus a we can write x square. So in the denominator of this function that is in place of x square we can write a plus b by 2 into x into into 2 upon this 2 upon a plus b and 2 upon b minus a outside the limit this is equal to 2 upon a plus b by 2 into 2 upon b minus b by 2 plus b by 2 x into 2x upon this is equal to b or you can say then into b minus a and this will upon 2 into 2 by this 4 tends to 0 plus b by 2 a plus b by 2 into x upon a plus b by 2x this into limit upon b minus a by 2 into a standard result of trigonometric limit which is limit x tends to 0 sin x upon x is equal to 1 this would result in this would be equal to now here else is 4 so this would be equal to a square upon 2 into 1 into 1 that is also 1 and this limit is also 1 b square minus a square upon that limit x tends to 0 is equal to b square minus a square upon c session so we have understood the solution of this question