 Hello and welcome to the session. In this session we shall discuss the L-Hopitase Rule. Till now we have studied that when x tends to a, then limit f of x by g of x is equal to limit f of x by limit g of x where limit of g of x is not equal to 0. As division by 0 is a meaningless operation, thus this rule of limits fail to give any information regarding the limit of a fraction where denominator tends to 0 are its limit. Now let us discuss some cases when denominator g of x tends to 0 as x tends to a. Now when g of x tends to 0 as x tends to a, then the denominator f of x may or may not tends to 0. Now if f of x does not tend to 0, then f of x by g of x cannot tend to any finite limit. Let if possible to a finite limit say L is equal to f of x by g of x into g of x. Now taking limits on both the sides, we have limit of f of x is equal to limit f of x by g of x into g of x which is equal to limit f of x by g of x into limit g of x which is equal to L into 0 as limit f of x tends to a finite limit by our supposition and this is equal to 0 that is limit f of x is equal to 0 thus we have a contradiction. Therefore three types of cases are possible. One fraction may tends to plus infinity. Second fraction may tends to minus infinity limit does not exist. For example limit 1 upon x raise to power 2 n as x tends to 0 is equal to plus infinity. If we put the value of x as 0 in this expression we get 1 upon 0 that is plus infinity. Second limit 1 upon x raise to power 2 n as x tends to 0 is equal to minus infinity. If we put the value of x as 0 in this expression we get 1 upon minus of 0 that is equal to minus infinity. Third limit 1 upon x cube as x tends to 0 does not exist as limit 1 upon x cube is equal to plus infinity as x tends to 0 plus and limit 1 upon x cube is equal to minus of infinity as x tends to 0 minus that is left hand limit is not equal to the right hand limit. Therefore the limit does not exist. Now we will discuss the indeterminate form. There are cases when the numerator of a fraction is also 0 or in other words we can say that a fraction where both numerator and denominator n to 0 as x tends to a is known as an indeterminate form that is 0 by 0 form when x is equal to a. Now let us discuss the rule called the L Hopkins rule. It uses derivatives and evaluates limits involving indeterminate forms converting it into determinate forms that is f of t and g of t and their derivatives are all continuous that x is equal to a is equal to g of a that is limit f of t as t tends to a is equal to limit g of t as t tends to a then limit f of t by g of t is equal to limit f dash of t by g dash of t as t tends to a provided g dash of t is not equal to 0. Now let us consider a curve passing through the origin and defined by the equations x is equal to f of t and x is equal to g of t. Let x y be the coordinates of a point p on the curve near the origin. Suppose a to be the value of t corresponding to the origin so that f of a is equal to 0 and g of a is equal to 0 also we have limit y upon x is equal to the limit tan of angle p on since tan of angle p on is given by perpendicular upon base that is pn upon on given by y by x which is equal to limit dy by dx as x tends to a hence limit f of t by g of t as t tends to a is equal to limit f dash of t by g dash of t as t tends to a where we do not differentiate f of t by g of t by question rule of differentiation but we differentiate the numerator and denominator separately if f dash of t and g dash of t is not of the form 0 by 0 when t is assigned value 0 then we have limit f of t by g of t as t tends to a is equal to f dash of t by g dash of t but if f dash of t by g dash of t is also of indeterminate form then we repeat the process limit f dash of t by g dash of t as t tends to a is equal to limit f double dash of t by g double dash of t as t tends to a and so on till we find fraction such that when value is substituted then both numerator and denominator do not vanish and hence we obtain a result for example we have limit sine of 2x by x as x tends to 0 now if we put the value of x as 0 in this we get sine of 0 by 0 which is a 0 by 0 form now applying n-hopitance rule we get limit differentiating sine of 2x with respect to x that is cos of 2x into differentiating 2x with respect to x that is 2 upon differentiating the denominator with respect to x that is differentiating x with respect to x that is 1 as x tends to 0 on putting the value of x as 0 we get cos of 0 into 2 which is equal to 1 into 2 since cos of 0 is 1 that is 1 into 2 that is 2 so the value of limit sine of 2x by x as x tends to 0 is 2 other indeterminate forms are infinity by infinity 0 into infinity infinity minus infinity 0 raise to power 0 infinity raise to power infinity raise to power 0 1 raise to power infinity let f of a be equal to infinity and g of a is equal to infinity so that f of a by g of a takes the form infinity by infinity then t approaches indefinitely near the value a now in such cases we write f of t by g of t is equal to 1 upon g of t by 1 upon f of t as 1 upon g of t is equal to 1 upon infinity that is 0 and 1 upon f of t is equal to 1 upon infinity that is 0 therefore we can consider this as taking the form 0 by 0 and can apply l-hopper's rule thus limit f of t by g of t to a is equal to limit 1 upon g of t whole upon 1 upon f of t as t tends to a differentiate numerator and denominator separately by applying questions rule of differentiation to both numerator and denominator but separately we get g of t into 0 minus 1 into g dash of t upon g of t the whole square whole upon f of t into 0 minus of 1 into f dash of t whole upon f of t the whole square therefore limit f of t upon g of t as t tends to a is equal to limit g dash of t upon f dash of t into f of t the whole square by g of t the whole square tends to a which can be written as limit f of t by g of t as t tends to a is equal to limit f of t by g of t the whole square as t tends to a into limit g dash of t by f dash of t as t tends to a which is equal to 1 is equal to limit f of t by g of t t tends to a into limit g dash of t by f dash of t as t tends to a which is equal to limit f of t by g of t as t tends to a is equal to limit f dash of t by g dash of t as t tends to a hence l-hopitals rule is also applicable to infinity by infinity form for example if we have limit log of x upon x cube as x tends to infinity now if we put the value of x as infinity in this expression we get log of infinity by infinity cube that is of infinity by infinity form so applying l-hopitals rule we get limit differentiating log of x with respect to x that is 1 upon x upon differentiating x cube with respect to x that is 3x square as x tends to infinity this can also be written as limit 1 upon 3x cube as x tends to infinity now if we put the value of x as infinity in this expression we get 0 now we shall discuss 0 into infinity form let f of a be 0 and g of a be infinity so that f of a into g of a takes the form 0 into infinity when t tends to a limit f of a into g of a when t tends to a is equal to limit f of a by 1 upon g of a when t tends to a since 1 upon g of a is equal to 1 upon infinity that is 0 therefore it will take the form 0 by 0 and we can apply l-hopitals rule for example if we have limit x square into log of x as x tends to 0 if we put the value of x as 0 in this expression we get 0 into log of 0 that is of 0 into infinity form since log of 0 is equal to minus infinity which can also be written as limit log of x upon 1 upon x square as x tends to 0 if we put the value of x as 0 we get log of 0 upon 1 upon 0 that is of infinity by infinity form now applying l-hopitals rule we get limit differentiating log of x with respect to x that is 1 upon x upon differentiating 1 upon x square with respect to x we get minus 2 upon x cube as x tends to 0 this can also be written as limit x square upon minus 2 is to 0 now if we put the value of x as 0 we get 0 upon minus 2 that is 0 this completes our session hope you enjoyed this session