Limits and choosing an epsilon properly?

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?

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arildno
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Gold Member
Dearly Missed
Is c=d?????????????

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

shmoe
Homework Helper
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?

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.

Last edited:
shmoe
Homework Helper
matrix_204 said:
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.

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 dont 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 existance.