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...
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...
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...
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...
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...