MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: R

There are no items in this cart.
Continue Shopping
Menu
sachin nayak Grade: 12
        

proof of L'hospitals theorem

6 years ago

Answers : (2)

SAGAR SINGH - IIT DELHI
879 Points
										

Dear sachin,


If f and g are differentiable in a neighborhood of x = c, and f(c) = g(c) = 0, then



provided the limit on the right exists. The same result holds for one-sided limits.


If f and g are differentiable and f(x) = g(x) = - then



provided the last limit exists.


Proof:


The first part can be proved easily, if the right hand limit equals f'(c) / g'(c): Since f(c) = g(c) = 0 we have




Taking the limit as x approaches c we get the first result. However, the actual result is somewhat more general, and we have to be slightly more careful. We will use a version of the Mean Value theorem:


Take any sequence {xn} converging to c from above. All assumptions of the generalized Mean Value theorem are satisfied (check !) on [c, xn]. Therefore, for each n there exists a number cn in the interval (c, xn) such that




We are all IITians and here to help you in your IIT JEE preparation.

All the best.


 If you like this answer please approve it....


win exciting gifts by answering the questions on Discussion Forum


 





Sagar Singh


B.Tech IIT Delhi





6 years ago
Karthik Eyan
45 Points
										

Consider the linear approximation to f(x) and g(x) at x=a:



The ratio of these for x near a is:



which, if g'(a) is not 0 approaches f '(a) / g'(a) as x approaches a.


If g'(a) = 0 and f '(a) = 0 we can apply the same rule to the derivatives, to give f "(a) / g"(a).


If these second derivatives are both 0 you can continue to higher derivatives, etc. the result will be the ratio of the first pair of non-vanishing higher derivatives at a.


Of course if the first non-vanishing derivative of the numerator is the kth and occurs before the kth then the ratio is 0; if the first non-vanishing entry of the denominator occurs after that of the numerator, the ratio goes to infinity at a.

6 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies
  • Complete JEE Main/Advanced Course and Test Series
  • OFFERED PRICE: R 15,000
  • View Details
Get extra R 3,000 off
USE CODE: CART20
Get extra R 440 off
USE CODE: CART20

Ask Experts

Have any Question? Ask Experts

Post Question

 
 
Answer ‘n’ Earn
Attractive Gift
Vouchers
To Win!!! Click Here for details