How to express the following limit in epsilon-delta language

In summary: In that case the definition is "Given any \epsilon> 0, there exist a \delta such that if 0< |x- a|< \delta then |f(x)- L|< \epsilon."
  • #1
Jacobpm64
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0
Express the following as an epsilon-delta proof (to show that it is continuous):

[tex]\lim_{x\rightarrow - \infty}f(x) = L[/tex]

Can I get some ideas on this one?
 
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  • #2
No one can do that question: you have failed to say what f is.
 
  • #3
Jacobpm64 said:
Express the following as an epsilon-delta proof (to show that it is continuous):

[tex]\lim_{x\rightarrow - \infty}f(x) = L[/tex]

Can I get some ideas on this one?
Just express that in terms of [itex]\epsilon[/itex], [itex]\delta[/itex] for general f and L? Then just use the definition of limit at infinity: the "standard" definition of limit, at some number a, is "Given any [itex]\epsilon> 0[/itex], there exist [itex]\delta[/itex] such that if [itex]|x- a|< \delta[/itex] then [itex]|f(x)- L|< \epsilon[/itex]." If you really mean "at infinity" then "[itex]|x- a|< \delta[/itex]" has to become "x sufficiently large" or "x> X" for some number X.

The definition, then, is
Given any [itex]\epsilon> 0[/itex], there exist a real number X such if x> X then [itex]|f(x)- L|< \epsilon[/itex].

The reason I said 'If you really mean "at infinity"' is that you also say "to show that it is continuous". You are taking the limit as x goes to infinity and a function is never continuous "at infinity". Taking the limit as x GOES to infinity simply means "as x gets larger without bound". "Infinity" is not a specific number and f is not defined "at infinity".
I wonder if you don't actually mean "limit as x goes to a" for some number a.
 

1. What is epsilon-delta language?

Epsilon-delta language is a mathematical notation used to express the concept of a limit in calculus. It is a formal way of defining the behavior of a function as the input approaches a certain value.

2. How is a limit expressed in epsilon-delta language?

In epsilon-delta language, a limit is expressed as "the limit of f(x) as x approaches a is L". This can also be written as "f(x) approaches L as x approaches a" or "the limit as x goes to a of f(x) is L".

3. What is the purpose of using epsilon-delta language?

The purpose of using epsilon-delta language is to provide a precise and rigorous definition of a limit. It allows for a clear understanding of how a function behaves near a specific point and enables the proof of limit properties and theorems.

4. How do you use epsilon-delta language to prove a limit?

To prove a limit using epsilon-delta language, one must show that for any small positive value of epsilon (ε), there exists a corresponding positive value of delta (δ) such that if the distance between x and a is less than δ, then the distance between f(x) and L is less than ε. This can be written as |x-a| < δ → |f(x)-L| < ε.

5. Can you provide an example of expressing a limit in epsilon-delta language?

For the function f(x) = 2x+3, the limit as x approaches 2 is 7. In epsilon-delta language, this can be written as: for any ε > 0, there exists a δ > 0 such that if |x-2| < δ, then |2x+3-7| < ε. This can also be written as lim x→2 (2x+3) = 7.

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