Limits and choosing an epsilon properly?

[matrix_204]

I really need help on solving this question:

Let d and K be given real numbers. Suppose that lim f(x) > K.
x->c
Show that there is a number h>0 such that f(x) > K for all x in the punctured open interval of width 2h centred at d.

The only hint that i was given was that if there are two real numbers as close as you like, then they are basically the same real.
How can i show this using this idea?

 

[arildno]

Is c=d?

 

[matrix_204]

yes that is the idea, but how am i suppose to show that it's equal

 

[shmoe]

Suppose \lim_{x\rightarrow c}f(x)=L>K

Write down the definition of the limit in this case. There's a |f(x)-L|<\epsilon part. This controls how close f is to L. By chosing epsilon properly, you can force f to be some distance away from any number not equal to L (on some punctured disc centered at c of course).

For example if you know |f(x)-L|<1/2 on some interval, then can f(x)=L+1/2 on this interval? Can it equal anything larger? What's the lowest it could be?

 

[matrix_204]

What does it mean when asking to chosing an epsilon properly?



Also, can someone clarify, how the graph of two horizontal lines work? i.e, y=L , y=f.

 

[shmoe]

What does it mean when asking to chosing an epsilon properly?

Also, can someone clarify, how the graph of two horizontal lines work? i.e, y=L , y=f.

You can think of epsilon as bounds for your function on the corresponding interval 0<|x-c|<\delta.On this interval, your function will only take on values above L-\epsilon and below L+\epsilon. How do you pick epsilon to leave K out of this range? If epsilon is too large, you won't be able to rule out the possibility that f(x)=K.


Your horizontal lines..y=L would just be a horizontal line at height L, y=f won't necessarily be a horizontal line, f is a function. I don't think I understand your question.

 

[matrix_204]

So in order to show that f(x)> K for all x.. what are the main steps required in proving this, and also is there any techniques that can be used in solving delta-epsilon type of problems? I m really confused in these types of problems, i don't seem to understand the concept of delta and epsilon, like i kno that they are really small and can be regarded as equal, or not equal but very close, etc.

 

[shmoe]

1) pick an \epsilon >0 so that if |f(x)-L|<\epsilon you know |f(x)-K|>0. Your particular epsilon will depend on how far L is from K.
2) Appeal to the definition of the limit to produce a \delta >0 so that if 0<|x-c|<\delta then |f(x)-L|<\epsilon. You won't know what this delta is, but the fact that the limit is L guarantees it's existence.
3) Combine the above.

epsilon-delta type proofs are important if you want an understanding of calculus. Everything you do later on depends on limits, and without understanding epsilon-delta you won't rigorously understand what a limit is. You really need to work some examples and try to understand the definition of the limit as much as possible. Practice is important.