Prove limits using epsilon delta definition

In summary, the conversation is about proving the continuity of the sine and cosine functions using the epsilon-delta definition. The poster is struggling to understand how to solve the problem and is asking for examples or clarification from their teacher. They discuss different methods for proving the continuity of the trig functions and the importance of understanding their definitions. Ultimately, the conversation ends with a suggestion to use trigonometric identities to prove continuity.
  • #1
fire sword
4
0

Homework Statement


http://store2.up-00.com/Sep12/JB498124.jpg [Broken]


2. The attempt at a solution
No attempts because i can't understand how to solve it
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
HOW have you been given definitions of the trig functions??
 
  • #3
the teacher want it by epsilon delta def only
 
  • #4
if u have examples for these types of problems u could provide me with it
and i will try to understand
 
  • #5
There is no epsilon definition of the sine and cosine functions.
I asked you how you have been defined THOSE functions?
Have they been proven, or defined as, continuous, for example?
Have you been given them as by representations of solutions of eigen value problems, or as exaples of power series?
 
  • #6
he only explained delta-epsilon definition for normal functions and gave us these to think how to prove them
 
  • #7
Well, then ASSUME that he regards that sine and cosine are CONTINUOUS functions (and that that is, therefore, allowable for you, as a student).
Make this an EXPLICIT assumption if you make a detailed reply.
Which of the limits are you thereby able to prove, given that assumption?
 
  • #8
Well, we really can't help you until you give us a definition of sine and cosine...
 
  • #9
You could do this without any definitions for sin and cos.

Are you allowed to use the mean value theorem yet is my only question? If so then these are all possible to do using ε-δ.

Otherwise the squeeze theorem would be the only other plausible way.
 
  • #10
Zondrina said:
You could do this without any definitions for sin and cos.

How can you possibly show that sin is continuous without defining sin?
 
  • #11
micromass said:
How can you possibly show that sin is continuous without defining sin?

I suppose I see what you're saying here ^, but continuity of sin alone over ℝ would be sufficient enough to show that |sinx - sina| < ε

I would assume the continuity of sin was trivial though?
 
  • #12
Zondrina said:
I suppose I see what you're saying here ^, but continuity of sin alone over ℝ would be sufficient enough to show that |sinx - sina| < ε

I would assume the continuity of sin was trivial though?

The first question requires him to show continuity of sine. So I don't think he can assume that.
 
  • #13
micromass said:
The first question requires him to show continuity of sine. So I don't think he can assume that.

Hmm that is a problem then. If you're not even given the fact that sin is continuous or any info about sin ( let alone the mean value theorem which is much later in any calculus course ), I don't think it would be possible to show this...

According to the OP, the professor threw a curve ball. More info would be needed I guess.
 
  • #14
micromass said:
The first question requires him to show continuity of sine. So I don't think he can assume that.

Nope.
It might be the question-for-the-Dummies:
Merely REFER to the general definition of continuity in order to show that you understand what it actually means that a continuous function is..continuous.

Given the next questions, though, OP has simply forgotten what he has told in lecture about the BEHAVIOUR of the trig functions.
 
  • #15
I hope that I'm not mistaken, but just start with the fact that [itex] |sin(x)|\leq|x|[/itex] for all real numbers.

then prove that: [itex]|sin(x)-sin(a)| \leq |x-a|[/itex]
you can also prove that [itex] |cos(x)-cos(a)| \leq |x-a|[/itex]

That tells you why sin(x) and cos(x) is continuous. This solves the first and second problems.

But it's still very hard to prove all of those limits with epsilon-delta definition. also for the last two ones you can't write an epsilon-delta definition, because the limit tends to go to infinity.
 
  • #16
If you define [itex]\cos(\theta)[/itex] and [itex]\sin(\theta)[/itex] in the usual geometric way, as the x- and y- coordinates of a point at angle [itex]\theta[/itex] on the unit circle, and you're willing to accept a simple geometric argument that these coordinates must approach (1,0) as [itex]\theta[/itex] approaches 0, then this means that [itex]\cos[/itex] and [itex]\sin[/itex] are continuous at 0. From that, you can use trigonometric identities to get continuity elsewhere, for example
[tex]\sin(x+h) - \sin(x) = \sin(x)\cos(h) + \cos(x)\sin(h) - \sin(x)[/tex]
so
[tex]\begin{align*}
\lim_{h \rightarrow 0} (\sin(x+h) - \sin(x)) &= \sin(x) \left(\lim_{h \rightarrow 0}\cos(h)\right) + \cos(x)\left(\lim_{h \rightarrow 0}\sin(h)\right) - \sin(x) \\
&= \sin(x) \cos(0) + \cos(x) \sin(0) - \sin(x) \\
&= \sin(x) + 0 - \sin(x) = 0
\end{align*}[/tex]
and therefore [itex]\sin[/itex] is continuous at [itex]x[/itex]. I'm not sure how you would turn this into a fully rigorous [itex]\epsilon-\delta[/itex] argument, however.
 

What is the epsilon-delta definition of a limit?

The epsilon-delta definition of a limit is a mathematical approach used to formally prove the existence of a limit. It involves selecting an arbitrary small value (epsilon) and finding a corresponding value (delta) within the domain of the function that will guarantee the output (y) will be within epsilon of the limit (L) for all inputs (x).

Why is the epsilon-delta definition important?

This definition is important because it allows us to prove the existence of a limit with mathematical rigor. It also helps us understand how a function behaves near a specific input and provides a way to quantify the concept of "close enough" when dealing with limits.

How do I use the epsilon-delta definition to prove a limit?

To prove a limit using the epsilon-delta definition, you need to start by selecting an arbitrary epsilon value. Then, using algebraic manipulation, you can express the distance between the limit and the output in terms of the distance between the input and the delta value. Finally, you can choose a delta value that satisfies the given conditions and proves the existence of the limit.

What are the common mistakes when using the epsilon-delta definition to prove a limit?

Some common mistakes when using this definition include not selecting an arbitrary epsilon value, not manipulating the expression correctly, and not choosing a delta value that satisfies the given conditions. It is also important to remember that this definition only proves the existence of a limit, not the actual value of the limit.

Can I use the epsilon-delta definition for all types of functions?

Yes, the epsilon-delta definition can be used for all types of functions, including continuous, discontinuous, and piecewise functions. However, the approach may differ depending on the type of function and the conditions given for the limit.

Similar threads

  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
946
  • Calculus and Beyond Homework Help
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
425
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
22
Views
3K
  • Calculus and Beyond Homework Help
Replies
26
Views
2K
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
4K
Back
Top