I've been tutoring calculus related subjects for 6+ years. I will share some common tips and tricks on how to solve Limits, Derivatives, and Integral questions. Hope it can make Calc easier for you!
If you have questions about a specific problem, feel free to email me at rebecca.k12math@gmail.com.
you're a genius, i tested by substituting the value of x to 10,000 integer and its came close to value of -2 , so basically any limits related to infinity, with any variables will only involve the power degree of the D and N from the expression, and other constants can be omitted, pure genius :) i love it
its very easy problem this problem is solved by studdents of 10 11th class in india we can solve this by differentiation also 1st comment please pin my comment
... A clear presentation on an indeterminate limit form ... just an accidental small spelling error at the beginning ... 'indeterminate' instead of 'inderminate', but of course you know this Rebecca ... thanks and best wishes, Jan-W
The Exponential Function e^x is realised from a binomial in two variables and has nothing to do with limit theory: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-J3itUB3wGmQ.html
Not necessarily but sort of. It depends on what you want to take for your starting point. Some people define ln(x)=int_1^x (1/t)dt, then you can show the properties of ln from there. Afterwards, you can then define the inverse function of ln and show it must be an exponential. You get the idea. It depends on which way you want to go, and historically logs were more familiar first and used first before exponentials because of their uses in cartography, navigation, etc. One could also argue that this limit is how one defines e to begin with. Depending on how you want to set things up again, most people are familiar being introduced to e via this limit and are given this to be the definition.
Theory says that if lim x --> a f(x) equals b, then lim x --> a (f(x))^2 equals b^2. So we can square f to get a simplified function of the form p(x)/q(x) where p, q are polynomials. This should trigger you, as polynomials simplify to constants when taking the derivative. So you can just start applying L'hôpital's rule however many times you need and doing this will give 4. Since this 4 represents the b^2, our answer is sqrt(4) = +/- 2. Sign analysis shows that in f, if x goes to infintiy we get inf/(-inf), so the -2 is the correct answer. Hence lim x --> infinity of f(x) = -2 Its an amazing way of getting rid of square roots. *Edit why the fk is my text crossed out
Actually, in this special case you can rewrite the numerator as (x^2 - 1) + 1. Since the first term has x - 1 as a factor, you are left with x + 1 after division and a left over fraction 1/(x - 1). I agree that long division is the ordinary,ordinary approach with these integrals.
L'Hopital's should only be chosen in cases where there is no alternative. This is the standard method. Using L'Hopital's rule is much faster but that's not the standard way of approach.
Thank you Thank you Thank you!!!!!!!! Now I see you simplify the top but not the bottom to cancel out the top in method one. Thank you!!!!!!!!!!!!!!!!!!!!
