QUESTION-URGENT-Epsilon-Delta Proof of Limits Being Equal

[NOAM]

HI,
How would I go about proving that two limits are equal to each other using the Epsilon-Delta definition?
Moreover how can I prove that:

[FONT=MathJax_Main]lim[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]→[FONT=MathJax_Main]0[FONT=MathJax_Math-italic]f[FONT=MathJax_Main]([FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]lim[FONT=MathJax_Math-italic]x[FONT=MathJax_Main]→[FONT=MathJax_Math-italic]a[/FONT][/FONT][/FONT][/FONT][FONT=MathJax_Math-italic]f[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]a[/FONT][FONT=MathJax_Main])[/FONT][/FONT][/FONT][/FONT][/FONT]limx→0f(x)=limx→af(x−a)[/FONT][/FONT]​

using the Epsilon-Delta definition? The intuition for this seem clear. However, I have do not know how a formal proof can be developed.



Why do I need to assume that δ=δ0?

 

[Dr.Peterson]

HI,
How would I go about proving that two limits are equal to each other using the Epsilon-Delta definition?
Moreover how can I prove that:

[FONT=MathJax_Main]lim_x→0f(x) = lim_x→af(x−a)[/FONT]
​

using the Epsilon-Delta definition? The intuition for this seem clear. However, I have do not know how a formal proof can be developed.



Why do I need to assume that δ=δ0?

Your question suggests that you have seen a proof and are asking why one step is needed. If so, please show it. If not, please explain what you mean by the question in bold.

Also, you appear to have copied and pasted from a source that duplicates material and loses some formatting. Above, I have tried to fix that; be sure to preview posts to make sure they are readable.

As for how I would prove this, I would start by stating what the definition of the first limit implies (that for any epsilon, ...), and then use that fact to show that the second limit exists and has the same value. I can't really say more without seeing your attempt.

 

[mmm4444bot]

Please take a few minutes to familiarize yourself with the forum's guidelines. You may start with this summary. Thank you! :cool:

 

[pka]

HI,
How would I go about proving that two limits are equal to each other using the Epsilon-Delta definition?
Moreover how can I prove that:
limx→0f(x)=limx→af(x−a)limx→0f(x)=limx→af(x−a)
using the Epsilon-Delta definition? The intuition for this seem clear. However, I have do not know how a formal proof can be developed.
Why do I need to assume that δ=δ0?

I completely agree with the other two replies. Now that said, here are my own concerns.
This is well known problem in courses such as "the theory of calculus" or whatever an equivalent might be named.
Prove \(\displaystyle \mathop {\lim }\limits_{x \to 0} f(x) = \mathop {\lim }\limits_{x \to a} f(x - a)\). Here is the deal: there are as many proofs for this as there are people who have written on it. AND they all may be different.
NOAM, the mistake you made is posting a step in someone else's proof.
You should have posted the proof and then asked a question about the step giving you pause. Please do that.