# lim

1. ### Lim n^(-1cos1)

So I've wolframed it and it shows that lim n^(-1cos1) is equal to zero. In fact infinity over negative number between 0 and 1 is equal to zero, why is that?
2. ### A property of the lim sup

Hello :), I want to prove that if X_{j(n)} is a subsequence of X_n , \lim \sup X_{j(n)} \leq \lim \sup X_n Is intuitively trivial but I am still trying to prove it. Thanks for your attention.
3. ### lim of x ( (e^2x) -1 )

the ans given is infinity , but i gt 0 here's my working lim of x ( (e^2x) -1 ) = ∞ (( e^2/∞ ) -1 )= ∞ (1-1) = -∞ (0) which is correct ?
4. ### lim (cos x )^(1/x ) when x = 0

here's my working , is it correct ?
5. ### lim

a)2 b)2 c)1 d)1 right or not?
6. ### lim (cot x ) / ln x when x approaches 0

Is my ans correct ? i dont have the ans btw.
7. ### Lim / Derivatives

got got f(x) = 2cos(x) and a=pi/3 but got it wrong.
8. ### Given lim h -> 0, how to prove: ln(1 + hr) / h = r ?

Is it possible to prove it using derivative first principle? I know how to prove lim n->max+ [ 1 + 1/n ]^n = e but I am not sure how to prove the captioned one. thank you very much.
9. ### Finding & Proving the limit of lim (x,y)->(0,0) sin(x-y)/cos(x+y)

It says in my textbook that sin(x-y)/cos(x+y) = sin(0)/cos(0) = 0/1 = 0, so 0 is the limit. For most of the other problems in the section, I have to show that when the limit exists using the definition of limits of two-variable functions (that sqrt(x^2 + y^2) < delta implies that |f(x,y)-L| <...
10. ### lim sup and lim inf of a countable collection of sets

Hi all, Problem statement: To build a case of a countable collection of sets of real numbers for which the lim inf and lim sup are not equal. My Attempt: (I normally associate lim sup and lim inf with sequences so I'm finding it a bit hard to think in terms of sets (intervals)) Anyway, let...
11. ### Interesting result of lim k—»0

I discovered this interesting result while working on another problem. \lim_{k \to 0}\frac{e^{kx}-kx-1}{e^k-k-1}=\ ? Can you solve it? I have the answer; please PM me if you'd like to see it. I'll post it here in a few days. Good luck! Cheers, ~ Justin
12. ### lim of function combining ln and trigonometry

is there a method (arithmetic simplification or something) which with i can solve lim that combine ln, e, and trigonometry? for example: lim_{x\rightarrow0}\frac{ln(cosx)}{cosx-1} and: lim_{x\rightarrow\infty}\frac{2x-sin2x}{x^{2}+cos^{^{2}}x}
13. ### Solving lim x->∞ e^x cos x

Hi everybody :) I'm trying to solve this limit \lim_{x\to\infty}e^x cos(x) Since \lim_{x\to\infty}e^x = \infty and \lim_{x\to\infty}cos(x) is a Real number in [-1,1] is it correct to say that the solution is \infty? Thank you for your help :)
14. ### Proof that lim sup (sin n) = 1

Very similar to the idea that lim sup (sin x) = 1, but now we are dealing with just the natural numbers, not all the reals. I have started this proof but will need help completing it. Part A: Show lim sup (sin x) <= 1. 1. Observe, N \subset \Re. So \limsup (\sin n) \leq \limsup (\sin x). 2...
15. ### Proof that lim sup (sin x) = 1

Please tell me if the following is sufficient to prove that lim sup (sin x) = 1. I have marked steps that I might need to explain--i.e., prove as well--with a **. My question is, at an Analysis level, would this proof suffice? Part A: Show lim sup (sin x) >= 1. **1. Observe that sin(2*n*pi + x)...
16. ### Proof that lim x-> inf (x ^ (1/x)) = 1

I started this proof by letting f(x) = (x ^ (1/x)) - 1 and attempting to show that that converges to zero. It's pretty straightforward to establish that a limit exists--x > e^1 => f(x) > 0 and f'(x) < 0, so f is bounded below and decreasing. Now what? Should I try lim inf, knowing that the...
17. ### why didn't i solve the lim well?

\lim_{x\rightarrow-\infty}(\sqrt{x^{2}+x}-x) = ? . this is how i solve this, but i know it's not true.. why? (\sqrt{x^{2}+x}-x)=\frac{(\sqrt{x^{2}+x}-x)\cdot(\sqrt{x^{2}+x}+x)}{(\sqrt{x^{2}+x}+x)}=\frac{x}{\sqrt{x^{2}+x}+x}...
18. ### lim (1+1/x)^x

I need a proof that lim(1+1/x)^x=e. We defined that e=lim(1+1/n)^n and I need to prove that it's same for x (real number). Thanks.
19. ### Prove that lim sup(x_n) = max(lim sup(y_n), lim sup(z_n))

Hi there, my first question here. Hope someone can give me some hints on it. Thanks! Let (x_{n}) be a bounded sequence. For each n \in \mathbb{N}, let y_{n}=x_{2n} and z_{n}=x_{2n-1}. Prove that \lim \sup {x_n} = \max (\lim \sup {y_n},\lim \sup {z_n})
20. ### sups and lim

f is a continous function in (0,1) which maintains lim x-->0+ f(x)=-1, and lim x-->1- f(x)=1 let's define: A={x in (0,1)| f(x)=0} s=supA. i need to prove that f(s)=0. well, i know that in oter words i need to prove that s is part of the group A, and is its maximum. But i don;t know how to...