Homework Statement
Prove that if f is uniformly continuous on [a,b] and on [a,c] implies that f is uniformly continuous on [a,c].
Homework Equations
The Attempt at a Solution
This is my rough idea for a proof, can someone help be say this more formally? Is my thinking even...
lim x->0- f(x)=L implies that there exists a real number L s.t. epsilon>0 there exists delta>0 s.t. lf(x)-Ll<epsilon provided 0<a-x<delta.
I'm am getting confused with all these definitions though, can you help me organize the argument using the definitions?
So lim x->0+ f(x)=L implies there exists a real number L s.t. epsilon>0 there exists delta>0 s.t. lf(x)-Ll<epsilon provided 0<x-a<delta.
Then lim x->0+ f(-x)=L implies that there exists a real number L s.t. epsilon>0 there exists delta>0 s.t. lf(-x)-Ll<epsilon provided 0<x-a<delta.
I have...
Homework Statement
Prove that if f: R->R is an even function, then lim x->0 f(x)=L if and only if lim x->0+ f(x)=L.
Homework Equations
The Attempt at a Solution
So far I have:
If f is an even function f(x)=f(-x) for x in domain of f.
Then I am trying to apply the limit...
Homework Statement
Evaluate lim x-->infinity [x]/x and lim--> -infinity [x]/x.
Homework Equations
The Attempt at a Solution
The think the limits for both of these are 1. I also know that [x] is the largest integer not greater than x.
I think that I can use the squeeze theorm...
I don't know how to simplify it at all. My thought was to maybe to get common denominator or and then multiply by the conjugate, but I don't know if this is correct. I got l[2-(2)^(1/2) *((x^2)+1)^(1/2)]/(2((x^2)+1)^(1/2)l. I'm sorry, it's hard to type it here, does any of that make sense?
Homework Statement
Prove lim x--> -1
1/(sqrt((x^2)+1)
using epsilon, delta definition of a limit
Homework Equations
The Attempt at a Solution
I know that the limit =(sqrt(2))/2
And my proof is like this so far. Let epsilon >0 be given. We need to find delta>0 s.t. if...
Homework Statement
Suppose that u, v ∈ Z and (u,v) = 1. If u | n and v | n, show that uv | n. Show that this is false if (u,v) ≠ 1.
Homework Equations
a | b if b=ac
[b]3. The Attempt at a Solution
I understand this putting in numbers for u,v, and n but I don't know how to...
I'm sorry, I still don't understand how to write the proof for this divergent sequence. I have a similar problem done correctly I believe. Can someone help me write a proof like the following one, but for this sequence above??
Here is a similar lim, I think I have correctly proved...
Homework Statement
Prove that the given sequence diverges to infinity.
{an} = (-n^4+n^3+n)/(2n+7)
Homework Equations
Diverges definition
The Attempt at a Solution
So far I have:
Let M>0 and let N= something.
I'm having a hard time figuring out what N should equal for the...
I'm not sure how to write the reverse. Would I start with: lim n--->infinity(a_n -A)=0. So given epsilon>0, there exists N>0 s.t. la_n-Al<epsilon for all n>N.
I don't know if this is correct, and I don't know where to go after that.
I'm not good at writing proofs. So far I have:
Let {an} converge to A. Given epsilon>0, there exists N>0 s.t. lan-Al<epsilon for all n>N.
So l((a_n)-A)l<epsilon for all n>n.
Thus, we can write lim n--->infinity (a_n-A)=0.
Then, I'm not sure how to prove the statement's converse. Can...
I have already canceled the factor of n, and I am stuck at the next step. I have n/(n-1)(n-2)!
Any suggestions what to do next? How do I proof that =0?
Homework Statement
Prove that the sequence {a_n} converges to A if and only if lim n--->∞ (a_n-A)=0.
Homework Equations
The Attempt at a Solution
It's an if and only if proof, but I'm not sure how to prove it. Please help!
Homework Statement
Determine whether the given limit exists and find their values. Give clear explanations using limit properties.
Homework Equations
lim n--->∞ (n^2)/n!
The Attempt at a Solution
I know that the limit is 0, but I don't know how to show it in detailed steps...