Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
proof of L'hospitals theorem
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.
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
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
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.
Get your questions answered by the expert for free
You will get reply from our expert in sometime.
We will notify you when Our expert answers your question. To View your Question
Win Gift vouchers upto Rs 500/-
Register Yourself for a FREE Demo Class by Top IITians & Medical Experts Today !