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...