Evalute the following given in the picture.
Topic - Calculus
Level - Hard
Class - 11th , 12th or Graduates.
Answers
Ans--
0
Explanation--
First i did, integration by parts to solve the integration. Which took many steps and lengthy but it was worthy. Then i used simple L'Hospital rule in lim to solve the question.
Answer:
Step-by-step explanation:
Definite integration:-
Here, at first we substitute n as 1/t,where t will tend to 0,as n tends to infinity.
After this substitution,we get a 0/0 form.
So,we apply L hospital rule,by differentiating both numerator and denominator to remove the indeterminate form.
In numerator,a definite integral is there.
So,we apply Newton Leibnitz rule for antiderivative of definite integral.
We assume y=t*2018^t
Applying log both sides,we get an indeterminate form again which is at power of e.
Again applying L hospital rule,we get y=1
But,there was a denominator of 2(y/2) if you observe carefully.
Hence required limit is 1/2.