 Hi and how are you all today? The question says evaluate limit x approaches 0, x tan 4x upon 1 minus cos 4x. Now let us quickly solve the given question. Now here rewriting the given limit. Once again we have now it equal to limit x approaches 0, x tan 4x upon Now here we can write 1 minus cos 4x as 2 sin square 2x isn't it? Because 1 minus cos 2x or let's take it as theta here is equal to 2 sin square. Now this is further equal to limit x approaches 0. Now we are separating out our numerator in this form. We have tan 4x upon 4x so we will multiply it by 4x also into 4x square upon sin square 2x into 1 upon 4x. So we have approaches 0 tan 4x upon 4x. We will get cancelled out with this 4x square so we have 1 by 2 into. Now we can write this whole term as the whole square and separating out the limit we have 1 by 2 limit when x approaches to 0 then 4x also approaches to 0. So we have limit 4x approaching 0 tan 4x upon limit when x approaches 0 then 2x also approaches to 0. So we have limit 2x approaching 0 2x upon sin 2x into this is whole square so again we have limit 2x approaches 0 2x upon. Now in using the limit we have 1 by 2 this whole term will become equal to exactly 1 into 1 into 1. This is because limit x approaches to 0 the value of tan x upon x is equal to limit x approaches to 0 sin x upon x is equal to 1. So we have the answer to this question as that is answer to this limit as 1 by. So this completes the session hope you understood it well and enjoyed it too have a nice day.