 Hello and welcome to the session. In this session, we are going to discuss the following question which says that evaluate limit tan of x raised to pi quarter of x as x tends to pi by 2. We know that n-heptase rule states that if f of x and g of x are the two functions such that f of a is equal to 0, g of a is equal to 0, then limit x tends to a f of x upon g of x is equal to limit x tends to a f dash of x upon g dash of x. This rule is also applicable for f of a is equal to infinity, g of a is equal to infinity. With this key idea, let us proceed with the solution. We are to find the value of the expression limit x tends to pi by 2 tan of x raised to power quarter of x. Now, if we put the value of x as pi by 2, in this expression we get tan of pi by 2 raised to power quarter of pi by 2. Since tan of pi by 2 is not defined and quarter of pi by 2 is 0, therefore it takes infinity raised to power 0 form. Now, let y be equal to tan of x raised to power quarter of x. Now taking log on both these sides, we get log of y is equal to quarter of x into log of tan of x, which can be written as log of y is equal to log of tan of x by tan of x since quarter of x is equal to 1 upon tan of x. Now taking limit on both these sides, we have limit x tends to pi by 2 log of y is equal to limit x tends to pi by 2 log of tan of x upon tan of x. Now if we put the value of x as pi by 2, we get log of tan of pi by 2 upon tan of pi by 2. Since tan of pi by 2 is not defined, therefore it takes infinity by infinity form. Now according to L-Hopital's rule we have, if f of x and g of x are the functions such that f of a is equal to 0 and g of a is equal to 0, then limit x tends to a f of x upon g of x is equal to limit x tends to a f dash of x upon g dash of x. And this rule is also applicable for f of a is equal to infinity and g of a is equal to infinity. Now since the expression on the right hand side of the equation takes infinity by infinity form, therefore applying L-Hopital's rule, we have limit x tends to pi by 2 log of y is equal to limit x tends to pi by 2 differential of the numerator with respect to x and differential of the denominator with respect to x that is differentiating log of tan of x with respect to x. We have 1 upon tan of x into differential of tan of x with respect to x that is 6 square of x whole upon differential of tan of x with respect to x that is 6 square of x which is equal to limit x tends to pi by 2 1 upon tan of x that is limit x tends to pi by 2 quarter of x which is equal to quarter of pi by 2 that is 0. Therefore limit x tends to pi by 2 log of y is equal to 0 which can also be written as log of limit x tends to pi by 2 y is equal to 0 which implies that limit x tends to pi by 2 y is equal to e raised to power 0 which is equal to 1. So we have limit x tends to pi by 2 y is equal to 1 and we have assumed the value of y as tan of x raised to power quarter of x therefore limit x tends to pi by 2 y that is tan of x raised to power quarter of x is equal to 1 which is the required answer.