Let $\,a>0\,,\,a\neq1\,$ be a real number. We can prove by using the continuity of $\ln n$ function that $\;\lim\limits_{n\to\infty}\dfrac{\log_an}n=0\;$
However, this problem appears in my problems book quite early right after the definition of $\epsilon$-language definition of limit of a...
Let ##\varepsilon > 0## be arbitrary. Now define ##\delta = \text{min}\{\frac{a}{2}, \varepsilon \sqrt{a}\}##. Now since ##a>0##, we can deduce that ##\delta > 0##. Now assume the following
$$ 0< |x-a| < \delta $$
From this, it follows that ##0 < |x-a| < \frac{a}{2} ## and ##0 < |x-a| <...
This question consists of two parts: preliminary and the main question. Reading only the main question may be enough to get my point, but if you want details please have a look at the preliminary.
PRELIMINARY:
Let potential due to a small volume ##\delta## at a point ##(1,2,3)## inside it be...
I understand the concept of Epsilon-Delta proofs, but I can't understand why we have to do them.
What's the advantage of using this proof over just showing that the limit from the function approaches from the left and right are the same?
This is a simple exercise from Spivak and I would like to make sure that my proof is sufficient as the proof given by Spivak is much longer and more elaborate.
Homework Statement
Prove that \lim_{x\to a} f(x) = \lim_{h\to 0} f(a + h)
Homework EquationsThe Attempt at a Solution
By the...
Homework Statement
Hey guys. I am having a little trouble answering this question. I am teaching myself calc 3 and am a little confused here (and thus can't ask a teacher). I need to find the limit as (x,y) approaches (0,1) of f(x,y) when f(x,y)=(xy-x)/(x^2+y^2-2y+1).
Homework Equations...
I have only encountered questions that f(x)-L that can be factorize to get a constant, and delta is epsilon divide that number, as a high school student.
I have no idea how to choose a epsilon for this question.
Thanks.
he definition of the limit of a function is as follows:[5]
Let be a function defined on a subset https://upload.wikimedia.org/math/a/1/b/a1b67abab803e714098f3e69a33900da.png, let be a limit point of https://upload.wikimedia.org/math/f/6/2/f623e75af30e62bbd73d6df5b50bb7b5.png, and let be a...
Homework Statement
Use the epsilon delta definition to show that lim(x,y) -> (0,0) (x*y^3)/(x^2 + 2y^2) = 0
Homework Equations
sqrt(x^2) = |x| <= sqrt(x^2+y^2) ==> |x|/sqrt(x^2+y^2) <= 1 ==> |x|/(x^2+2y^2)?
The Attempt at a Solution
This limit is true IFF for all values of epsilon > 0, there...
Will epsilon delta test fail if curve changed direction within the +/- delta of the limit point?
Is there a scenario where no matter how small we pick delta to be, the frequency of the graph changing directions is always going to be a higher than delta's distance? In that scenario the limit...
I am struggling to properly understand the \varepsilon-\delta definition of limits.
So, f(x) gets closer to L as x approaches a. That is okay. However, taking the leap from there to the \varepsilon-\delta definition is something I have never really been able to do.
Why is the formulation we...
The question asks to proof that the limit given in incorrect by contradiction. I tried working using the estimation method and the weird thing is that I completed the proof and found that the supposedly "incorrect" limit gave a correct answer although it was supposed to give me a contradiction...
< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >
This is from the question list at the UC Davis Website epsilon delta exercise list.
In the exercise list we have:
Prove that
Which concludes with:
Thus, if , it follows that...
Hi
I'm new to limits and calculus in general. Our professor told us there existed some rigorous proof for a limit, but it was "beyond the scope of the course". All we needed to know about a limit was that (1)$$\lim_{x\to a} f(x)$$ is true iff when x approaches a from both directions p(x)...
Using epsilon delta, prove
$$\lim_{{n}\to{\infty}}\frac{2^n}{n!}=0$$
Doesn't seem too difficult, but I have forgotten how to do it. Obvious starting point is $\forall \epsilon >0$, $\exists N$ such that whenever $n>N,\left|\frac{2^n}{n!} \right|<\epsilon$.
I am currently having some issue understanding, what you may find trivial, epsilon-delta proofs. I have worked through Apostol Vol.1 and ran through Spivak and I found Apostol just uses neighborhoods in proofs instead of the epsilon-delta approach, while nesting neighborhoods is 'acceptable' I...
$$\lim_{{x}\to{2}}\frac{1}{x}=\frac{1}{2}$$
Here is what I have so far:
For all $\delta >0$, there exists an $x$ such that $0<|x-2|<\delta $, $|\frac{1}{x}-\frac{1}{2}<\epsilon$
Cut to the chase:
$$\frac{|x-2|}{|2x|}<\epsilon$$
I need to bound $\frac{1}{|2x|}$ somehow, and represent it with...
https://answers.yahoo.com/question/index?qid=20130915100124AAK4JAQ
I do not understand how they got:
"1 = |(1 plus d/2 - L) - (d/2 - L)| <= |1 plus d/2 - L| plus |d/2 - L| < 1/4 plus 1/4 = 1/2, "
Shouldn't it be $|(1+ \frac{\delta}{2} -L) + (\frac{\delta}{2} -L)|$, not $|(1+ \frac{\delta}{2}...
Hey there, I'm new to this forum. Today I thought I would brush up on my calculus.
I would just like to know if my method is correct. Is there an easier way to prove this ?
By the way, it's my first time using LaTeX, so bear with me.
I am trying to prove the following :
\lim_{x\rightarrow...
I seem to be having trouble with multivariable epsilon-delta limit proofs. I don't have a very good intuition for how \epsilon relates to \delta.
For example:
Prove \lim_{(x,y) \to (0,0)}\frac{2xy^2}{x^2+y^2} = 0
There are probably many ways to do this, but my teacher does it a certain way...
Homework Statement
Prove that if
##\left |x-x_{0} \right | < \frac{\varepsilon }{2}## and ##\left |y-y_{0} \right | < \frac{\varepsilon }{2}##
then
##|(x+y)-(x_0+y_0)| < \varepsilon ## and ##|(x-y)-(x_0-y_0)| < \varepsilon ##Homework Equations
Postulate and proof with real numbers as well...
I've been reading through Spivak's calculus, and the problem is the answer key i have a hold of is for a different edition so it often doesn't answer the correct questions.
Anyways, here they are:
Chapter 5 problem 10
b. Prove that lim x-> 0 f(x) = lim x-> a f(x-a)
c. Prove that lim...
This isn't really homework; It's just something that has been bothering me ever since I first learned calculus because I suck at epsilon-delta proofs.
Homework Statement
Show that 1/x is continuous at x=1
Homework Equations
If |x-a|<δ
Then |f(x)-f(a)|<ε
The Attempt at a Solution...
Homework Statement
When constructing an Epsilon Delta proof, why do we need to make a stipulation? For example, in most proofs for limits of quadratic functions, it is stipulated, for example, that δ≤1. Why is this needed anyway?
This is my thought process for a quadratic:
Prove that lim(x...
I started learning how to do these things today and boy, they take some interesting logic. Anyway, here's my attempt at one:
prove that the limit as (x,y) → (0,0) of [(x^2)(siny)^2]/(x^2 + 2y^2) exists
Here's what I did:
0<√(x^2 + y^2) < δ, |[(x^2)(siny)^2]/(x^2 + 2y^2) - 0| < ε...
Homework Statement
lim (x,y) -> (0,0) xy/sqrt(x^2+y^2) = 0
The Attempt at a Solution
my understanding of my actual goal here is kind of poor
given ε>0 there exist ∂>0 s.t. 0 < sqrt(x^2 + y^2) < ∂ then 0<|f(x,y) - L| < ε
| xy/sqrt(x^2 + y^2) - 0 | < ε
(xy * sqrt(x^2 + y^2)) /...
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
I want to show that \lim_{x \rightarrow 0}\frac{1}{x} does not exist by negating epsilon-delta definition of limit.
Homework Equations
The Attempt at a Solution
We say limit exists when:
\forall \epsilon > 0, \exists \delta > 0 : \forall x(0< \left| x\right| < \delta...
Homework Statement
Lim x→a of f(x) = c (Where c is a constant)
Homework Equations
The Attempt at a Solution
I have no idea. I am able to do these if I can manipulate fx-L to equal x-a but I am having trouble with this one. Please help me!
Homework Statement
Determine the limit l for a given a and prove that it is the limit by showing how to find δ such that |f(x)-l|<ε for all x satisfying 0<|x-a|<δ.
f(x)=x^{4}+\frac{1}{x}, a=1.
Homework Equations
I claim that \lim\limits_{x\rightarrow 1}x^{4}+\frac{1}{x}=2.
The...
Homework Statement
Determine the limit l for a given a and prove that it is the limit by showing how to find δ such that |f(x)-l|<ε for all x satisfying 0<|x-a|<δ.
f(x)=x^{2}, arbitrary a.Homework Equations
I will incorporate the triangle inequality in this proof.The Attempt at a Solution
We...
Homework Statement
lim 3 as x->6
lim -1 as x->2
Homework Equations
In the first weeks of a calculus class and doing these epsilon delta proofs.
As i am looking at two of the problems i have been assigned:
Lim 3 as x->6
Lim -1 as x->2
The Attempt at a Solution...
So far, all I understand is that the definition proves that there's a value of f(x,y) as f(x,y) approaches (x0,y0) which is sufficiently close to but not exactly the value at f(x0,y0). I am probably completely off... but I just don't understand the purpose of proving this. I also don't...
Homework Statement
I just want to make sure I include all the steps in doing this:
lim (6x-7) = 11
x->3
Homework Equations
The Attempt at a Solution
given ε>0, we need to find a δ>0, such that 0< lx-3l < δ then 0 < l (6x-7)-11 l < ε
To prove this I need to make 0 < l...
For part A, (described here: http://www.cramster.com/solution/solution/1157440) I don't understand why they say |x-2| < 1 and why \delta = min{1,ε/5}
In case you can't view the page:
lim x2+2x-5 = 3, x \rightarrow 2
Let ε > 0 and L = 3.
|x2 + 2x -5 -3| < ε
|x2 + 2x - 8| < ε
|x+4||x-2| < ε...
Homework Statement
if |x-3| < ε/7 and 0 < x ≤ 7 prove that |x^2 - 9| < ε
Homework Equations
The Attempt at a Solution
So ths is what I did so far.
|x+3|*|x-3| < ε (factored out the |x^2 - 9|)
|x+3|*|x-3| < |x+3|* ε/7 < ε (used the fact that |x-3| < ε/7)
|x+3|* ε/7 *7 <...
I'm sick of still not getting this. I bombed the epsilon delta part of my mid term. A site where it gives many problems on epsilon delta and solutions would be amazing.
Need help proving lim(x)->(a) sqrt(x)=sqrt(a) using epsilon delta definition.
Homework Statement
Prove that the limit of \sqrt{x} is \sqrt{a} as x approaches a
if a>0
Homework Equations
in words
By the epsilon delta definition we know that the distance between f(x) and the limit...
Homework Statement This is my first delt/epsilon proof ever, so please understand if I seem ignorant.
e=epsilon
d = delta
Let f(x) = 1/x for x>0
If e is any positive quantity, find a positive number d, which is such that:
if 0 < |x-2| < d, then |f(x) - 1/2| < e
Homework...
Homework Statement
Suppose |f(x)-5|<0.1 when 0<x<5.
Find all values \delta>0 such that |f(x)-5|<0.1 whenever 0<|x-2|<\delta
Homework Equations
The Attempt at a Solution
I know that 0<|x-2|<\delta
2-\delta<x<2+\delta
\delta=2
but how does this part of the equation help me find...
Hello,
I have stumbled upon a couple of proofs, but I can not seem to get an intuitive grasp on the what's and the whys in the steps of the proofs. Strictly logical I think I get it. Enough talk however.
Number 1.
"Let f be a continuous function on the real numbers. Then the set {x in R ...
Homework Statement
Prove the function f(x)= { 4 if x=0; x^2 otherwise
is discontinuous at 0 using epsilon delta.
Homework Equations
definiton of discontinuity in this case:
there exists an e>0 such that for all d>0 if |x-0|<d, |x^2-4|>e
The Attempt at a Solution
I'm confused...
Homework Statement
find an open interval about x0 on which the inequality lf(x)-Ll<\epsilon holds. Then give a value for \delta>0 such that for all x satisfying 0<lx-xol<\delta the inequality lf(x)-Ll<\epsilon holds
f(x)=x2
L=3
x0=-2
\epsilon=0.5
Homework Equations
The Attempt...
Recently in adv calc we have been dealing with the epsilon delta definition for continuity, and my professor said that it is ok to assume that delta<1. I actually used this to show that x^4 satisfies the epsilon delta condition but I'm not quite sure why we can take delta<1. I am sure you guys...
I have a problem on a take-home test, so I can't ask about the specific problem. So this is just going to be a general, how do I put stuff together problem.
I have a function of x and y that maps R2 into R1. The limit as (x,y)->(0,0) is zero, and I've worked through the various paths already...
Homework Statement
Is my understanding correct?
As delta(y) and delta(x) approach X from points to the left and points to the right of X (x is what we wish to find the derivative of) then the x and y values of points to the left and right approach the x and y values of X.
And as the...
Homework Statement
prove that the lim as x goes to 4 of x^2 + x -11 = 9
This is the example used on Paul's Online Notes on limits in calculus which can be found here http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx (I really like this resource.)
Homework Equations
Paul...