Prove Limit Rule: Learn the Constant Concept

[shihab-kol]

Hello, I would like to begin by saying that this does not fall into any homework or course work for me. It is just my interest.
I need to prove that limit of a constant gives the constant it self. Can some one provide a link? I have exams or I would have searched myself but unfortunately I don't have time. So, I had to post this thread.

 

[jedishrfu]

Check these notes, it came up as the very first search choice using "proof of limit rule":

http://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx

Please try to search this stuff in future before asking for help.

 

[Mark44]

Hello, I would like to begin by saying that this does not fall into any homework or course work for me. It is just my interest.

This is one of the easiest of limits to prove, using the definition of a limit to prove it. If f(x) = k, a constant, can you show that $|f(x) - f(a)| < \epsilon$ when $|x - a| < \delta$?

 

[shihab-kol]

This is one of the easiest of limits to prove, using the definition of a limit to prove it. If f(x) = k, a constant, can you show that $|f(x) - f(a)| < \epsilon$ when $|x - a| < \delta$?

If I take f(a) =k for some interval around x , then |f(x) - f(a)| =0 <E since by definition E >0.
But will this work?

 

[Mark44]

If I take f(a) =k for some interval around x , then |f(x) - f(a)| =0 <E since by definition E >0.
But will this work?

It will work, but you need to say more.
Someone gives you a value of $\epsilon > 0$. Can you specify a number $\delta > 0$ so that when $|x - a| < \delta$, then $|f(x) - f(a)| < \epsilon$?