I have got some HW problems based on limits and continuity...please help in solving those problems..
Q1) Find the limit of the following using L'Hospital rule....
a) lim x-> 0+ [sin(x) - x] / [e^x - 1]
b) lim x-> pi/2 [pi/2 - x] . tan(x)
Q2) Prove using the Mean value theorem: (x-1)/x < ln(x) < x-1, when x>1
Edit: duplicated mathstuds fine work, sorry
L'Hopital's rule states that if and or and
Go for it. Question 1 is easy.
For question 2 start by taking the derivative of all parts of the inequality.
For the 2nd rewrite it as
now as this is of the form
Using L.H again we get
For the 2nd question consider the function
f is continous on and differentiable on so it satisfies the hypothesis of the MVT
by the MVT where
Let a=1 and b=x and we get
simplifying we get
Also by the same reasoning so
so finally we get
Yes Sir! I got what you were trying to do after using the L'hospital Rule but I was asking that why are you manipulating the term? Since, in the problem it was given as ( pi/2 - x) sinx / cosx...but you have mentioned that we can rewrite the function as
So why are you changing the sign here? And I guess that will change the answer. I got 1 from the function given in the original question and you are getting -1 from this one.
I am sorry to bother you. But I am just a little bit confused and want to learn if something I do not know.
Thanks so much once again!
The original problem was this
This doesnt quite fit the form to use L'hospitals rule.
We need to manipulate the equation into the form or
Hence I changed
Now this is of the form
So from here we take the derivative of the numerator and the denomiator as per L.H's rule.
I guess I don't understand what you mean changing the sign?